Algebraic number theory From Wikipedia, the free encyclopedia
Contents 1
2
3
4
5
Abel’s irreducibility theorem
1
1.1
References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Abelian extension
2
2.1
2
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abhyankar’s inequality
3
3.1
3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abhyankar’s lemma
4
4.1
4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abstract analytic number theory
5
5.1
Arithmetic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5.1.1
Additive number systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
5.3
Methods and techniques
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Arithmetical formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
5.4
See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
5.5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
5.3.1
6
7
Additive polynomial
8
6.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
6.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
6.3
The ring of additive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
6.4
The fundamental theorem of additive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . .
9
6.5
See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
6.6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
6.7
External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Adele ring
10
7.1
Definitions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
7.2
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
7.3
Idele group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
i
ii
8
9
CONTENTS 7.4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
7.5
See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
7.6
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
7.7
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Adelic algebraic group
13
8.1
Ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
8.2
Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
8.3
History of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
8.4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
8.5
External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Adjunction (field theory)
15
9.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
9.2
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
9.3
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
9.4
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
9.5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
10 Albert–Brauer–Hasse–Noether theorem
17
10.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
10.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
10.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
10.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
10.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
11 Algebraic closure
19
11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
11.2 Existence of an algebraic closure and splitting fields . . . . . . . . . . . . . . . . . . . . . . . . .
19
11.3 Separable closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
11.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
11.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
12 Algebraic extension
21
12.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
12.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
12.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
12.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
12.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
13 Algebraic function field
23
13.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
13.2 Category structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
13.3 Function fields arising from varieties, curves and Riemann surfaces . . . . . . . . . . . . . . . . . .
23
CONTENTS
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13.4 Number fields and finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
13.5 Field of constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
13.6 Valuations and places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
13.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
13.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
14 Algebraic number field
25
14.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
14.1.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
14.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
14.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
14.3 Algebraicity and ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
14.3.1 Unique factorization and class number . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
14.3.2 ζ-functions, L-functions and class number formula . . . . . . . . . . . . . . . . . . . . . .
27
14.4 Bases for number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
14.4.1 Integral basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
14.4.2 Power basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
14.5 Regular representation, trace and determinant
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
14.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
14.6 Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
14.6.1 Archimedean places
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
14.6.2 Nonarchimedean or ultrametric places . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
14.6.3 Prime ideals in OF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
14.7 Ramification
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
14.7.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
14.7.2 Dedekind discriminant theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
14.8 Galois groups and Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
14.9 Local-global principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
14.9.1 Local and global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
14.9.2 Hasse principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
14.9.3 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
14.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
14.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
14.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
15 Algebraic number theory
36
15.1 History of algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
15.1.1 Diophantus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
15.1.2 Fermat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
15.1.3 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
15.1.4 Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
15.1.5 Dedekind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
iv
CONTENTS 15.1.6 Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
15.1.7 Artin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
15.1.8 Modern theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
15.2 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
15.2.1 Unique factorization and the ideal class group . . . . . . . . . . . . . . . . . . . . . . . .
38
15.2.2 Factoring prime ideals in extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
15.2.3 Primes and places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
15.2.4 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
15.2.5 Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
15.3 Major results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
15.3.1 Finiteness of the class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
15.3.2 Dirichlet’s unit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
15.3.3 Reciprocity laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
15.3.4 Class number formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
15.4 Related areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
15.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
15.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
15.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
15.7.1 Introductory texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
15.7.2 Intermediate texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
15.7.3 Graduate level accounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
15.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
16 Algebraically closed field
44
16.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
16.2 Equivalent properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
16.2.1 The only irreducible polynomials are those of degree one . . . . . . . . . . . . . . . . . .
44
16.2.2 Every polynomial is a product of first degree polynomials . . . . . . . . . . . . . . . . . .
44
16.2.3 Polynomials of prime degree have roots . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
16.2.4 The field has no proper algebraic extension . . . . . . . . . . . . . . . . . . . . . . . . . .
45
16.2.5 The field has no proper finite extension . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
n
16.2.6 Every endomorphism of F has some eigenvector . . . . . . . . . . . . . . . . . . . . . .
45
16.2.7 Decomposition of rational expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
16.2.8 Relatively prime polynomials and roots . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
16.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
16.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
16.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
17 All one polynomial
47
17.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
17.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
17.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
CONTENTS
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17.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Archimedean property
48 49
18.1 History and origin of the name of the Archimedean property
. . . . . . . . . . . . . . . . . . . .
50
18.2 Definition for linearly ordered groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
18.2.1 Ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
18.3 Definition for normed fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
18.4 Examples and non-examples
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
18.4.1 Archimedean property of the real numbers . . . . . . . . . . . . . . . . . . . . . . . . . .
51
18.4.2 Non-Archimedean ordered field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
18.4.3 Non-Archimedean valued fields
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
18.4.4 Equivalent definitions of Archimedean ordered field . . . . . . . . . . . . . . . . . . . . .
52
18.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
18.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
19 Arithmetic and geometric Frobenius 19.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Arithmetic dynamics
54 54 55
20.1 Definitions and notation from discrete dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
20.2 Number theoretic properties of preperiodic points . . . . . . . . . . . . . . . . . . . . . . . . . .
55
20.3 Integer points in orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
20.4 Dynamically defined points lying on subvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
20.5 p-adic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
20.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
20.7 Other areas in which number theory and dynamics interact . . . . . . . . . . . . . . . . . . . . . .
56
20.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
20.9 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
20.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
20.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
21 Artin L-function
59
21.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
21.2 Functional equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
21.3 The Artin conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
21.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
21.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
21.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
22 Artin reciprocity law
62
22.1 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
22.2 Finite extensions of global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
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CONTENTS 22.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
22.2.2 Relation to quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
22.3 Cohomological interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
22.4 Alternative statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
22.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
22.6 References
65
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23 Artin transfer (group theory)
66
23.1 Transversals of a subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
23.2 Permutation representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
23.3 Artin transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
23.3.1 Independence of the transversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
23.3.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
23.3.3 Wreath product of H and S(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
23.3.4 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
23.3.5 Wreath product of S(m) and S(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
23.3.6 Cycle decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
23.3.7 Normal subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
23.4 Computational implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
23.4.1 Abelianization of type (p,p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2
23.4.2 Abelianization of type (p ,p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
23.5 Transfer kernels and targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
23.6 Abelianization of type (p,p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
23.7 Abelianization of type (p2 ,p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
23.7.1 First layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
23.7.2 Second layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
23.7.3 Transfer kernel type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
23.7.4 Connections between layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
23.8 Inheritance from quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
23.8.1 Passing through the abelianization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
23.8.2 TTT singulets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
23.8.3 TKT singulets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
23.8.4 TTT and TKT multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
23.8.5 Inherited automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
23.9 Stabilization criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
23.10Structured descendant trees (SDTs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
23.11Pattern recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
23.11.1 Historical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
23.12Commutator calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
23.13Systematic library of SDTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
23.13.1 Coclass 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
23.13.2 Coclass 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
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23.13.3 Coclass 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
23.14Arithmetical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
23.14.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
23.14.2 Comparison of various primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
23.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
24 Artin’s conjecture on primitive roots
96
24.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
24.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
24.3 Proof attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
24.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
24.5 References
97
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25 Biquadratic field
98
25.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Brauer group
98 99
26.1 Construction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
26.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
26.3 Brauer group and class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 26.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 26.5 General theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
26.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 26.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 26.8 References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
26.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 26.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 27 Brauer–Wall group
103
27.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 27.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 27.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 27.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 28 Brumer–Stark conjecture
105
28.1 Statement of the conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 28.2 Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 28.3 Function field analogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 28.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 29 Carlitz exponential
107
29.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 29.2 Relation to the Carlitz module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 29.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
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30 Characteristic (algebra) 30.1 Other equivalent characterizations
109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
30.2 Case of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 30.3 Case of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 30.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 31 Class field theory
112
31.1 Formulation in contemporary language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 31.2 Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 31.3 Generalizations of class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 31.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 32 Class formation
116
32.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 32.2 Examples of class formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 32.3 The first inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 32.4 The second inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 32.5 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 32.6 Tate’s theorem and the Artin map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 32.7 The Takagi existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 32.8 Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 32.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 32.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 33 Class number formula
122
33.1 General statement of the class number formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 33.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 33.3 Dirichlet class number formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 33.4 Galois extensions of the rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 33.5 Abelian extensions of the rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 33.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 33.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 34 Class number problem
125
34.1 Gauss’s original conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 34.2 Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 34.3 Lists of discriminants of class number 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 34.4 Modern developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 34.5 Real quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 34.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 34.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 34.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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34.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 35 CM-field
128
35.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 35.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 35.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 35.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 36 Compatible system of ℓ-adic representations
130
36.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 36.2 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 36.3 Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 36.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 36.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 37 Complete field
131
37.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 38 Complex multiplication
132
38.1 Example of the imaginary quadratic field extension . . . . . . . . . . . . . . . . . . . . . . . . . . 132 38.2 Abstract theory of endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 38.3 Kronecker and abelian extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 38.4 Sample consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 38.5 Singular moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 38.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 38.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 38.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 38.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 39 Composite field (mathematics)
137
39.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 40 Conductor (class field theory)
138
40.1 Local conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 40.1.1 More general fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 40.1.2 Archimedean fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 40.2 Global conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 40.2.1 Algebraic number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 40.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 40.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 41 Conductor of an abelian variety
141
41.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 41.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
x
CONTENTS 41.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
42 Conductor-discriminant formula
143
42.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 42.2 Example
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
42.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 42.4 References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
43 Conjugate element (field theory)
145
43.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 43.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 43.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 43.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 44 Cubic field
147
44.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 44.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 44.3 Galois closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 44.4 Associated quadratic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 44.5 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 44.6 Unit group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 44.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 44.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 45 Cubic reciprocity
152
45.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 45.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 45.2.1 Primes ≡ 1 (mod 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 45.2.2 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 45.2.3 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 45.2.4 Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 45.2.5 Other theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 45.3 Eisenstein integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 45.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 45.3.2 Facts and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 45.3.3 Cubic residue character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 45.3.4 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 45.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 45.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 45.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 45.6.1 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 45.6.2 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 45.6.3 Eisenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
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45.6.4 Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 45.6.5 Modern authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 45.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 46 Cyclotomic character
162
46.1 p-adic cyclotomic character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 46.2 As a compatible system of ℓ-adic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 46.3 Geometric realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 46.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 46.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 46.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 47 Cyclotomic field
164
47.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 47.2 Relation with regular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 47.3 Relation with Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 47.3.1 List of Class Numbers to Cyclotomic Field
. . . . . . . . . . . . . . . . . . . . . . . . . 165
47.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 47.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 47.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 48 Cyclotomic unit
167
48.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 48.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 49 Dedekind domain
169
49.1 The prehistory of Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 49.2 Alternative definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 49.3 Some examples of Dedekind domains 49.4 Fractional ideals and the class group
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
49.5 Finitely generated modules over a Dedekind domain . . . . . . . . . . . . . . . . . . . . . . . . . 172 49.6 Locally Dedekind rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 49.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 49.8 References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
49.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 49.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 50 Dedekind zeta function
174
50.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 50.1.1 Euler product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 50.1.2 Analytic continuation and functional equation . . . . . . . . . . . . . . . . . . . . . . . . 174 50.2 Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 50.3 Relations to other L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
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CONTENTS 50.4 Arithmetically equivalent fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 50.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 50.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
51 Degree of a field extension
177
51.1 Definition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 51.2 The multiplicativity formula for degrees
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
51.2.1 Proof of the multiplicativity formula in the finite case . . . . . . . . . . . . . . . . . . . . 178 51.2.2 Proof of the formula in the infinite case . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 51.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 51.4 Generalization 51.5 References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
52 Different ideal
180
52.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 52.2 Relative different . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 52.3 Ramification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 52.4 Local computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 52.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 52.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 53 Differential Galois theory
183
53.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 53.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 53.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 53.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 54 Discrete valuation
185
54.1 Discrete valuation rings and valuations on fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 54.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 54.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 55 Discriminant
187
55.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 55.2 Formulas for low degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 55.3 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 55.4 Quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 55.5 Discriminant of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 55.6 Nature of the roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 55.6.1 Quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 55.6.2 Cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 55.6.3 Higher degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 55.7 Discriminant of a polynomial over a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . 193
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55.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 55.8.1 Discriminant of a conic section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 55.8.2 Discriminant of a quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 55.8.3 Discriminant of an algebraic number field . . . . . . . . . . . . . . . . . . . . . . . . . . 195 55.9 Alternating polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 55.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 55.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 56 Discriminant of an algebraic number field
197
56.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 56.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 56.3 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 56.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 56.5
Relative discriminant . . . . . . . . . . . . . . . . . . . . . . . . 199 56.5.1 Ramification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 56.6 Root discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 56.7 Relation to other quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 56.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 56.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 56.9.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 56.9.2 Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 56.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 57 Drinfeld module
204
57.1 Drinfeld modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 57.1.1 The ring of additive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 57.1.2 Definition of Drinfeld modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 57.1.3 Examples of Drinfeld modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 57.2 Shtukas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 57.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 57.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 57.4.1 Drinfeld modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 57.4.2 Shtukas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 58 Dual basis in a field extension
207
59 Eisenstein reciprocity
208
59.1 Background and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 59.1.1 Primary numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 59.1.2 m-th power residue symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 59.2 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 59.2.1 First supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 59.2.2 Second supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
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CONTENTS 59.2.3 Eisenstein reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 59.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 59.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 59.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 59.5.1 First case of Fermat’s last theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 59.5.2 Powers mod most primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 59.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 59.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 59.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
60 Eisenstein sum
212
60.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 60.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 61 Eisenstein’s criterion
213
61.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 61.1.1 Cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 61.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 61.3 Basic proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 61.4 Advanced explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 61.5 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 61.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 61.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 61.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 62 Elementary number 62.1 References
218
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
63 Elliptic Gauss sum
219
63.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 63.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 64 Elliptic unit
221
64.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 64.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 65 Embedding problem
222
65.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 65.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 65.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 66 Equally spaced polynomial
224
66.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 66.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
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67.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 68 Euclidean field
226
68.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 68.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 68.3 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 68.4 Euclidean closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 68.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 68.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 69 Euler system
228
69.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 69.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 69.2.1 Cyclotomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 69.2.2 Gauss sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 69.2.3 Elliptic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 69.2.4 Heegner points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 69.2.5 Kato’s Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 69.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 69.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 69.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 70 Explicit reciprocity law
231
70.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 70.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 70.2.1 Archimedean fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 70.2.2 Unramified case: the tame Hilbert symbol . . . . . . . . . . . . . . . . . . . . . . . . . . 231 70.2.3 Ramified case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 70.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 70.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 70.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 70.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 71 Exponential field
234
71.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 71.2 Trivial exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 71.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 71.4 Exponential rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 71.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 71.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 71.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
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72 Exponentially closed field
237
72.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 72.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 72.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 72.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 73 Extension and contraction of ideals
239
73.1 Extension of an ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 73.2 Contraction of an ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 73.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 73.4 Extension of prime ideals in number theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
73.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 73.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 74 Field (mathematics)
241
74.1 Definition and illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 74.1.1 First example: rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 74.1.2 Second example: a field with four elements . . . . . . . . . . . . . . . . . . . . . . . . . . 243 74.1.3 Alternative axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 74.2 Related algebraic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 74.2.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 74.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 74.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 74.4.1 Rationals and algebraic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 74.4.2 Reals, complex numbers, and p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . 244 74.4.3 Constructible numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 74.4.4 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 74.4.5 Archimedean fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 74.4.6 Field of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 74.4.7 Local and global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 74.5 Some first theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 74.6 Constructing fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 74.6.1 Closure operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 74.6.2 Subfields and field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 74.6.3 Rings vs fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 74.6.4 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 74.7 Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 74.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 74.8.1 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 74.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 74.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 74.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
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74.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 74.13Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 74.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 75 Field extension
252
75.1 Definitions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
75.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 75.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 75.4 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 75.5 Algebraic and transcendental elements and extensions . . . . . . . . . . . . . . . . . . . . . . . . 253 75.6 Normal, separable and Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 75.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 75.8 Extension of scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 75.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 75.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 75.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 75.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 76 Field norm
256
76.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 76.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 76.3 Properties of the norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 76.4 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 76.5 Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 76.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 76.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 76.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 77 Field of fractions
259
77.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 77.2 Construction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
77.3 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 77.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 77.5 References 78 Field trace
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 261
78.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 78.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 78.3 Properties of the trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 78.4 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 78.4.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 78.5 Trace form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 78.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
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78.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 78.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 78.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 79 Finite extensions of local fields
265
79.1 Unramified extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 79.2 Totally ramified extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 79.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 79.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 80 Formal group
267
80.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 80.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 80.3 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 80.4 The logarithm of a commutative formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . 269 80.5 The formal group ring of a formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 80.6 Formal group laws as functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 80.7 The height of a formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 80.8 Lazard ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 80.9 Formal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 80.10Lubin–Tate formal group laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 80.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 80.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 81 Formally real field
274
81.1 Alternative Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 81.2 Real Closed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 81.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 81.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 82 Fractional ideal
276
82.1 Definition and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 82.2 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 82.3 Divisorial ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 82.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 82.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 82.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 83 Frobenius endomorphism
278
83.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 83.2 Fixed points of the Frobenius endomorphism 83.3 As a generator of Galois groups 83.4 Frobenius for schemes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
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83.4.1 The absolute Frobenius morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 83.4.2 Restriction and extension of scalars by Frobenius
. . . . . . . . . . . . . . . . . . . . . . 280
83.4.3 Relative Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 83.4.4 Arithmetic Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 83.4.5 Geometric Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 83.4.6 Arithmetic and geometric Frobenius as Galois actions . . . . . . . . . . . . . . . . . . . . 284 83.5 Frobenius for local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 83.6 Frobenius for global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 83.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 83.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 83.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 84 Function field sieve
287
84.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 85 Fundamental discriminant 85.1 Connection with quadratic fields
288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
85.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 85.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 85.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 86 Fundamental theorem of algebra
290
86.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 86.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 86.2.1 Complex-analytic proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 86.2.2 Topological proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 86.2.3 Algebraic proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 86.2.4 Geometric proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 86.3 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 86.4 Bounds on the zeros of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 86.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 86.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 86.6.1 Historic sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 86.6.2 Recent literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 86.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 87 Fundamental theorem of Galois theory
300
87.1 Explicit description of the correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 87.2 Properties of the correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 87.3 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 87.4 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 87.5 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 87.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
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CONTENTS 87.7 Infinite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 87.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
88 Fundamental unit (number theory)
304
88.1 Real quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 88.2 Cubic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 88.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 88.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 88.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 89 Galois cohomology 89.1 History
306
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
89.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 90 Galois extension
308
90.1 Characterization of Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 90.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 90.3 References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
90.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 91 Galois module
310
91.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 91.1.1 Ramification theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 91.2 Galois module structure of algebraic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 91.3 Galois representations in number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 91.3.1 Artin representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 91.3.2 ℓ-adic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 91.3.3 Mod ℓ representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 91.3.4 Local conditions on representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 91.4 Representations of the Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 91.4.1 Weil–Deligne representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 91.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 91.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 91.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 91.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 92 Generic polynomial
314
92.1 Groups with generic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 92.2 Examples of generic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 92.3 Generic Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 92.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 93 Genus character
316
93.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
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94 Genus field
317
94.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 94.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 95 Global field
318
95.1 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 95.2 Analogies between the two classes of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 95.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 95.3.1 Hasse-Minkowski theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 95.3.2 Artin reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 95.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 95.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 96 Glossary of field theory
321
96.1 Definition of a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 96.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 96.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 96.4 Types of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 96.5 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 96.6 Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 96.7 Extensions of Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 96.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 97 Golod–Shafarevich theorem
327
97.1 The inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 97.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 97.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 98 Grothendieck’s Galois theory
329
98.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 99 Ground field
331
99.1 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 99.2 In linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 99.2.1 In algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 99.2.2 In Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 99.2.3 In Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 99.2.4 In Diophantine geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 99.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 100Group cohomology 100.1Motivation
333
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
100.2Formal constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 100.2.1 Long exact sequence of cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
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CONTENTS 100.2.2 Cochain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 100.2.3 The functors Extn and formal definition of group cohomology . . . . . . . . . . . . . . . . 335 100.2.4 Group homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
100.3Functorial maps in terms of cochains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 100.3.1 Connecting homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 100.4Non-abelian group cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 100.5Connections with topological cohomology theories . . . . . . . . . . . . . . . . . . . . . . . . . . 337 100.6Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 100.6.1 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 100.6.2 H 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 100.6.3 H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 100.6.4 Change of group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 100.6.5 Cohomology of finite groups is torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 100.7History and relation to other fields
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
100.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 100.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 101Grunwald–Wang theorem 101.1History
341
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
101.2Wang’s counter-example
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
101.2.1 An element that is an nth power almost everywhere locally but not everywhere locally . . . . 341 101.2.2 An element that is an nth power everywhere locally but not globally . . . . . . . . . . . . . 342 101.3A consequence of Wang’s counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 101.4Special fields
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
101.5Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 101.6Explanation of Wang’s counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 101.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 101.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 101.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 102Hardy field
344
102.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 102.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 102.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 102.4In model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 102.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 103Hasse invariant of an algebra
346
103.1Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 103.2Global fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 103.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 103.4Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
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104Hasse norm theorem
348
104.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 104.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 105Hasse principle
349
105.1Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 105.2Forms representing 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 105.2.1 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 105.2.2 Cubic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 105.2.3 Forms of higher degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 105.3Albert–Brauer–Hasse–Noether theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 105.4Hasse principle for algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 105.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 105.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 105.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 105.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 106Heegner number
352
106.1Euler’s prime-generating polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 106.2Almost integers and Ramanujan’s constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 106.2.1 Detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 106.3Pi formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 106.4Other Heegner numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 106.5Consecutive primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 106.6Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 106.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 107Heegner point
357
107.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 108Herbrand quotient
358
108.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 108.1.1 Alternative definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 108.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 108.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 108.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 109Hermite’s problem
360
109.1Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 109.2Hermite’s question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 109.3Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 109.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 110Higher local field
362
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110.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 110.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 110.3Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 110.4Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 110.5Measure, integration and harmonic analysis on higher local fields . . . . . . . . . . . . . . . . . . . 363 110.6Class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 110.7Higher adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 110.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 110.9References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
110.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 111Hilbert class field
365
111.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 111.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 111.3Additional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 111.4Explicit constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 111.5Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 111.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 111.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 112Hilbert symbol
367
112.1Quadratic Hilbert symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 112.1.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 112.1.2 Interpretation as an algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 112.1.3 Hilbert symbols over the rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 112.1.4 Kaplansky radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 112.2The general Hilbert symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 112.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 112.2.2 Hilbert’s reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 112.2.3 Power residue symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 112.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 112.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 113Hilbert’s ninth problem 113.1Progress made
371
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
113.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 113.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 113.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 114Hilbert’s twelfth problem
372
114.1Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 114.2Modern development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 114.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
CONTENTS 114.4References
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115Hurwitz problem
374
115.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 115.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 116Hyper-finite field
376
116.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 117Hyperreal number
377
117.1The transfer principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 117.2Use in analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 117.2.1 Calculus with algebraic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 117.2.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 117.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 117.4Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 117.4.1 From Leibniz to Robinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 117.4.2 The ultrapower construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 117.4.3 An intuitive approach to the ultrapower construction . . . . . . . . . . . . . . . . . . . . . 380 117.5Properties of infinitesimal and infinite numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 117.6Hyperreal fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 117.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 117.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 117.9Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 117.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 118Ideal class group
384
118.1History and origin of the ideal class group
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
118.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 118.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 118.4Relation with the group of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 118.5Examples of ideal class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 118.5.1 Class numbers of quadratic fields 118.6Connections to class field theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
118.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 118.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 118.9References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
119Ideal norm
388
119.1Relative norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 119.2Absolute norm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
119.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 119.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
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120Integer
390
120.1Algebraic properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
120.2Order-theoretic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 120.3Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 120.4Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 120.5Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 120.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 120.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 120.8References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
120.9Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 120.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 121Isomorphism extension theorem 121.1Isomorphism extension theorem
396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
121.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 122Iwasawa theory
397
122.1Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 122.2Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 122.3Connections with p-adic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 122.4Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 122.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 122.6References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
122.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 122.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 123Jacobson–Bourbaki theorem
400
123.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 123.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 124Krasner’s lemma
402
124.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 124.2Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 124.3Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 124.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 124.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 125Kronecker–Weber theorem
404
125.1Field-theoretic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 125.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 125.3Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 125.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 126Kummer theory
406
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126.1Kummer extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 126.2Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 126.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 126.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 127Lafforgue’s theorem
409
127.1Langlands conjectures for GL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 127.2Representations of the Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 127.3Automorphic representations of GLn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 127.4Drinfeld’s theorem for GL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 127.5Lafforgue’s theorem for GLn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 127.6Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 127.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 127.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 127.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 128Langlands dual group
412
128.1Definition for separably closed fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 128.2Definition for groups over more general fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 128.3Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 128.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 129Langlands–Deligne local constant
414
129.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 129.2Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 129.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 129.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 130Lazard’s universal ring
416
130.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 131Leopoldt’s conjecture
417
131.1Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 131.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 132Levi-Civita field
419
132.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 132.2Extensions and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 132.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 132.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 133Linked field
421
133.1Linked quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 133.2Linked fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
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133.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 133.4Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 133.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 134Liouville’s theorem (differential algebra) 134.1Definitions
423
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
134.2Basic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 134.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 134.4Relationship with differential Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 134.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 134.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 135List of algebraic number theory topics
426
135.1Basic topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 135.2Important problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 135.3General aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 135.4Class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 135.5Iwasawa theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 135.6Arithmetic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 136List of number fields with class number one
429
136.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 136.2Quadratic number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 136.2.1 Real quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 136.2.2 Imaginary quadratic fields
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
136.3Cubic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 136.4Cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 136.5CM fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 136.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 136.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 136.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 137Local class field theory
432
137.1Connection to Galois groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 137.2Lubin–Tate theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 137.3Higher local class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 137.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 137.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 137.6Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 138Local Euler characteristic formula
434
138.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 138.1.1 Case of finite modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
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138.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 138.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 139Local field
436
139.1Induced absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 139.2Non-archimedean local field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 139.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 139.2.2 Higher unit groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 139.2.3 Structure of the unit group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 139.3Higher-dimensional local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 139.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 139.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 139.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 139.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 139.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 140Local Fields
440
140.1Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 140.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 141Local Langlands conjectures
441
141.1Local Langlands conjectures for GL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 141.2Representations of the Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 141.3Representations of GLn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 141.4Local Langlands conjectures for GL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 141.5Local Langlands conjectures for GL
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
141.6Local Langlands conjectures for other groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 141.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 141.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 142Lubin–Tate formal group law
445
142.1Definition of formal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 142.2Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 142.3Generating ramified extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 142.4Connection with stable homotopy theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 142.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 142.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 143Lüroth’s theorem
448
143.1Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 143.2Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 143.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 144Maillet’s determinant
449
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CONTENTS 144.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
145Minimal polynomial (field theory)
450
145.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 145.1.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 145.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 145.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 145.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
146Minkowski space (number field)
452
146.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 147Mode of a linear field
453
148Modulus (algebraic number theory)
454
148.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 148.2Ray class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 148.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 148.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 148.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 149Monogenic field
457
149.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 149.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 150Multiplicative group
458
150.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 150.2Group scheme of roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 150.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 150.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
150.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 151Nagata’s conjecture
460
151.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 152Narrow class group
461
152.1Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 152.2Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 152.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 152.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 152.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
153Newton polygon
463
153.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 153.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
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153.3Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 153.4Symmetric function explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 153.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 153.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 153.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 154Non-abelian class field theory
467
154.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 154.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 155Norm form
469
155.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 156Norm group
470
156.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 156.2References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
157Normal basis
471
157.1Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 157.2Primitive normal basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 157.3Free elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 157.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 157.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 158Normal extension
473
158.1Equivalent properties and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 158.2Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 158.3Normal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 158.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 158.5References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
159p-adic Hodge theory
475
159.1General classification of p-adic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 159.2Period rings and comparison isomorphisms in arithmetic geometry . . . . . . . . . . . . . . . . . . 475 159.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 159.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 159.4.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 159.4.2 Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 160p-adic number
479
160.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 160.2p-adic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 160.3Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 160.4Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
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CONTENTS 160.4.1 Analytic approach
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
160.4.2 Algebraic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 160.5Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 160.5.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 160.5.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 160.5.3 Metric completions and algebraic closures . . . . . . . . . . . . . . . . . . . . . . . . . . 486 160.5.4 Multiplicative group of Qp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 160.5.5 Analysis on Qp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 160.6Rational arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 160.7Generalizations and related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 160.8Local–global principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 160.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 160.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 160.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 160.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 161p-adic order
490
161.1Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 161.1.1 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 161.1.2 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 161.2p-adic Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 161.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 161.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 162p-adically closed field
493
162.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 162.2The Kochen operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 162.3First-order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 162.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
162.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 163Parshin chain
496
163.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 164Perfect field
497
164.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 164.2Field extension over a perfect field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 164.3Perfect closure and perfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 164.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 164.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 164.6References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
164.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
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165Perfectoid
500
165.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 165.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 165.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 166Polynomial basis
501
166.1Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 166.2Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 166.3Squaring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 166.4Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 166.5Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 166.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 166.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 167Power residue symbol
504
167.1Background and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 167.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 167.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 167.4Relation to the Hilbert symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 167.5Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 167.6Power reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 167.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 167.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 167.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 168Primary extension
507
168.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 168.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 169Primitive element (finite field)
508
169.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 169.1.1 Number of Primitive Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 169.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 169.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 169.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 170Primitive element theorem 170.1Terminology
509
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
170.2Existence statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 170.3Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 170.4Constructive results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 170.5Example
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
170.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
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170.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 170.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 171Primitive polynomial (field theory)
512
171.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 171.2Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 171.2.1 Field element representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 171.2.2 Pseudo-random bit generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 171.2.3 CRC codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 171.3Primitive trinomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 171.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 171.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 172Principalization (algebra)
514
172.1Extension of classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 172.2Artin’s reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 172.3Commutative diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 172.4Class field tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 172.5Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 172.6History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 172.6.1 Quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 172.6.2 Cubic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 172.6.3 Sextic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 172.6.4 Quartic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 172.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 172.8Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 172.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 173Profinite integer
521
173.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 173.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 173.3References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
173.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 174Proofs of Fermat’s theorem on sums of two squares
523
174.1Euler’s proof by infinite descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 174.2Lagrange’s proof through quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 174.3Dedekind’s two proofs using Gaussian integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 174.4Zagier’s “one-sentence proof” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 174.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 174.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 174.7External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
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175Proofs of quadratic reciprocity
528
175.1Proofs that are accessible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 175.2Eisenstein’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 175.2.1 Proof of Eisenstein’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 175.3Proof using algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 175.3.1 Cyclotomic field setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 175.3.2 The Frobenius automorphism
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
175.3.3 Completing the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 175.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
175.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 176Pseudo algebraically closed field
536
176.1Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 176.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 176.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 176.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
177Pseudo-finite field
538
177.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 178Purely inseparable extension
539
178.1Purely inseparable extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 178.1.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 178.2Galois correspondence for purely inseparable extensions . . . . . . . . . . . . . . . . . . . . . . . 540 178.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 178.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 179Pythagoras number
542
179.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 179.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 179.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 179.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 180Pythagorean field
544
180.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 180.1.1 Equivalent conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 180.2Models of geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 180.3Diller–Dress theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 180.4Superpythagorean fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 180.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 180.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 181Quadratic field
547
181.1Ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
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181.2Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 181.3Prime factorization into ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 181.4Quadratic subfields of cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 181.4.1 The quadratic subfield of the prime cyclotomic field . . . . . . . . . . . . . . . . . . . . . 548 181.4.2 Other cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 181.5Orders of quadratic number fields of small discriminant . . . . . . . . . . . . . . . . . . . . . . . 548 181.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 181.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 181.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 181.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 182Quadratic integer
550
182.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 182.2Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 182.3Norm and conjugation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
182.4Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 182.5Quadratic integer rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 182.5.1 Examples of complex quadratic integer rings
. . . . . . . . . . . . . . . . . . . . . . . . 552
182.5.2 Examples of real quadratic integer rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 182.5.3 Principal rings of quadratic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 182.5.4 Euclidean rings of quadratic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 182.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 182.7References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
183Quadratic reciprocity
556
183.1Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 183.2Terminology, data, and two statements of the theorem . . . . . . . . . . . . . . . . . . . . . . . . 557 183.2.1 Table of quadratic residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 183.2.2 −1 and the first supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 183.2.3 ±2 and the second supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 183.2.4 ±3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 183.2.5 ±5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 183.2.6 Gauss’s version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 183.2.7 Table of quadratic character of primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 183.2.8 Legendre’s version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 183.3Connection with cyclotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 183.4History and alternative statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 183.4.1 Fermat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 183.4.2 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 183.4.3 Legendre and his symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 183.4.4 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 183.4.5 Other statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
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183.4.6 Jacobi symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 183.4.7 Hilbert symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 183.5Other rings
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
183.5.1 Gaussian integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 183.5.2 Eisenstein integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 183.5.3 Imaginary quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 183.5.4 Polynomials over a finite field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 183.6Higher powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 183.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 183.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 183.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 183.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 184Quadratically closed field
571
184.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 184.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 184.3Quadratic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 184.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 184.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 185Quartic reciprocity
573
185.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 185.2Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 185.2.1 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 185.2.2 Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 185.2.3 Burde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 185.2.4 Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 185.3Gaussian integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 185.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 185.3.2 Facts and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 185.3.3 Quartic residue character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 185.3.4 Statements of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 185.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 185.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 185.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 185.7Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 185.7.1 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 185.7.2 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 185.7.3 Eisenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 185.7.4 Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 185.7.5 Modern authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 185.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
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186Quasi-algebraically closed field
584
186.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 186.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 186.3C fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 186.3.1 C 2 fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 186.3.2 Weakly Ck fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 186.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 186.5Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 186.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 187Quasi-finite field
588
187.1Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 187.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 187.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 187.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
188Quaternionic structure
590
188.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 189Ramification (mathematics)
591
189.1In complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 189.2In algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 189.3In algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 189.3.1 In algebraic extensions of Q
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
189.3.2 In local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 189.4In algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 189.5In algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 189.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 189.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 189.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 190Ramification group
594
190.1Ramification groups in lower numbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 190.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 190.2Ramification groups in upper numbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 190.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 190.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 190.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 191Ramification theory of valuations
598
191.1Galois case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 191.1.1 Decomposition group and inertia group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 191.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
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191.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 192Rational number
599
192.1Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 192.2Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 192.2.1 Embedding of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 192.2.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 192.2.3 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 192.2.4 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 192.2.5 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 192.2.6 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 192.2.7 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 192.2.8 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 192.2.9 Exponentiation to integer power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 192.3Continued fraction representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 192.4Formal construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 192.5Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 192.6Real numbers and topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 192.7p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 192.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 192.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 192.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 193Rational reciprocity law
606
193.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 194Rational variety
607
194.1Rationality and parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 194.2Rationality questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 194.3Lüroth’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 194.4Unirationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 194.5Rationally connected variety
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
194.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 194.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 194.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 195Ray class field
610
195.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 195.2Ray class fields using ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 195.3Ray class fields using ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 195.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 195.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
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196Real closed field
612
196.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 196.2Model theory: decidability and quantifier elimination . . . . . . . . . . . . . . . . . . . . . . . . . 613 196.3Order properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
196.4The generalized continuum hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 196.5Examples of real closed fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 196.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 196.7References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
196.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 197Reciprocity law
616
197.1Quadratic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 197.2Cubic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 197.3Quartic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 197.4Octic reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 197.5Eisenstein reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 197.6Kummer reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 197.7Hilbert reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 197.8Artin reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 197.9Local reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 197.10Explicit reciprocity laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 197.11Power reciprocity laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 197.12Rational reciprocity laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 197.13Scholz’s reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 197.14Shimura reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 197.15Weil reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 197.16Langlands reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 197.17Yamamoto’s reciprocity law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 197.18See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 197.19References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 198Regular extension
621
198.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 198.2Self-regular extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 198.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 199Regular prime
623
199.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 199.1.1 Class number criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 199.1.2 Kummer’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 199.2Siegel’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 199.3Irregular primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
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199.3.1 Infinitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 199.3.2 Irregular pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 199.3.3 Irregular index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 199.3.4 Euler irregular primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 199.3.5 Strong Irregular primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 199.3.6 Weak irregular primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 199.3.7 Harmonic irregular primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 199.4History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 199.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 199.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 199.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 199.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 200Resolvent (Galois theory)
629
200.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 200.2Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 200.3Resolvent method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 200.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
201Rigid analytic space
632
201.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 201.2Other formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 201.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 201.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 202Ring class field
634
202.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 202.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 203Ring of integers
635
203.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 203.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 203.3Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 203.4Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 203.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 203.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 204Fontaine’s period rings
637
204.1The ring B R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 204.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 204.2.1 Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 205Rupture field
638
205.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
xlii
CONTENTS 205.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 205.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
206S-unit
639
206.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 206.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 206.3S-unit equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 206.4References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
206.5Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 207Separable extension
641
207.1Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 207.2Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 207.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 207.4Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 207.5The definition of separable non-algebraic extension fields
. . . . . . . . . . . . . . . . . . . . . . 643
207.6Differential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 207.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 207.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 207.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 207.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 208Separable polynomial
646
208.1Older definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 208.2Separable field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 208.3Applications in Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 208.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 208.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 209Serre’s conjecture II (algebra)
648
209.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 209.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 210Simple extension
649
210.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 210.2Structure of simple extensions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
210.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 210.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 211Skolem–Mahler–Lech theorem
651
211.1Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 211.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 211.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
CONTENTS
xliii
212Splitting field
652
212.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 212.2Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 212.3Constructing splitting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 212.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 212.3.2 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 212.3.3 The field Ki[X]/(f(X)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 212.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 212.4.1 The complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 212.4.2 Cubic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 212.4.3 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 212.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 212.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 212.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 213Splitting of prime ideals in Galois extensions 213.1Definitions
656
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656
213.1.1 The Galois situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 213.1.2 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 213.2Example — the Gaussian integers 213.2.1 The prime p = 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
213.2.2 Primes p ≡ 1 mod 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
213.2.3 Primes p ≡ 3 mod 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
213.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 213.3Computing the factorisation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
213.3.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 213.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 213.5References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660
214Square class
661
214.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 215Stark conjectures
662
215.1Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 215.2Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 215.3Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 215.4Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 215.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 215.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 216Strassmann’s theorem
664
216.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 216.2Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664
xliv
CONTENTS 216.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 216.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664
217Stufe (algebra)
665
217.1Powers of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 217.2Positive characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 217.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 217.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 217.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 217.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 217.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 218Superreal number
667
218.1Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 218.2References 218.3Bibliography
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
219Supersingular prime (for an elliptic curve) 219.1References
668
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
220Symbol (number theory)
669
220.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 220.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 221Takagi existence theorem
671
221.1Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 221.2A well-defined correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 221.3Earlier work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 221.4History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 221.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 221.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 222Tate cohomology group
673
222.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 222.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 222.3Tate’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 222.4Tate-Farrell cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 222.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 222.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 223Tate duality
675
223.1Local Tate duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 223.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 223.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
CONTENTS 224Tate’s thesis
xlv 676
224.1Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 224.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 225Teichmüller character
678
225.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 225.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 226Teichmüller cocycle
679
226.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 226.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 227Tensor product of fields
680
227.1Compositum of fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 227.2The tensor product as ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 227.3Analysis of the ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 227.4Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 227.5Classical theory of real and complex embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 227.6Consequences for Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 227.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 227.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 227.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 227.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 228Thin set (Serre)
683
228.1Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 228.2Hilbertian fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 228.3WWA property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 228.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 229Timeline of class field theory
686
229.1Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 229.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 230Totally imaginary number field
688
230.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 231Totally real number field
689
231.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 231.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 232Tower of fields
691
232.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 232.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
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CONTENTS
233Transcendence degree
692
233.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 233.2Analogy with vector space dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 233.3Facts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
233.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 233.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 234Tschirnhaus transformation
694
234.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 234.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 235Tsen rank
695
235.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 235.2Norm form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 235.3Diophantine dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 235.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 235.5References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696
236Twisted polynomial ring
697
236.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 236.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 236.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 237u-invariant
699
237.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 237.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 237.3The general u-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 237.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 237.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 238Unique factorization domain
701
238.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 238.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 238.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 238.4Equivalent conditions for a ring to be a UFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 238.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 238.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 239Unit (ring theory)
705
239.1Group of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 239.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 239.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 240Universal quadratic form
707
CONTENTS
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240.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 240.2Forms over the rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 240.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 240.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 241Valuation (algebra)
709
241.1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 241.1.1 Associated objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 241.2Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 241.2.1 Equivalence of valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 241.2.2 Extension of valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 241.2.3 Complete valued fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 241.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 241.3.1 π-adic valuation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
241.3.2 P-adic valuation on a Dedekind domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 241.3.3 Geometric notion of contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 241.4Vector spaces over valuation fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 241.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 241.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 241.7References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
241.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 242Valuation ring
714
242.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 242.2Definitions 242.3Construction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
242.4Dominance and integral closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 242.5Ideals in valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 242.6Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 242.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 242.8References 243Weil group
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 719
243.1Weil group of a class formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 243.2Weil group of an archimedean local field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 243.3Weil group of a finite field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 243.4Weil group of a local field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 243.5Weil group of a function field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 243.6Weil group of a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 243.7Weil–Deligne group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 243.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 243.9Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
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243.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 244Zahlbericht
722
244.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 244.2Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 244.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 244.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 244.5Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 724 244.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 244.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 244.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
Chapter 1
Abel’s irreducibility theorem In mathematics, Abel’s irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel,[1] asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F.[2][3] Corollaries of the theorem include:[2] • If ƒ(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root √ with it. For example, x2 − 2 is irreducible over the rational numbers and has 2 as a root; hence there is no √ linear or constant polynomial over the rationals having 2 as a root. Furthermore, there is no same-degree polynomial that shares any roots with ƒ(x), other than constant multiples of ƒ(x). • If ƒ(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots.
1.1 References [1] Abel, N. H. (1829), “Mémoire sur une classe particulière d'équations résolubles algébriquement” [Note on a particular class of algebraically solvable equations], Journal für die reine und angewandte Mathematik 4: 131–156, doi:10.1515/crll.1829.4.131. [2] Dörrie, Heinrich (1965), 100 Great Problems of Elementary Mathematics: Their History and Solution, Courier Dover Publications, p. 120, ISBN 9780486613482. [3] This theorem, for minimal polynomials rather than irreducible polynomials more generally, is Lemma 4.1.3 of Cox (2012). Irreducible polynomials, divided by their leading coefficient, are minimal for their roots (Cox Proposition 4.1.5), and all minimal polynomials are irreducible, so Cox’s formulation is equivalent to Abel’s. Cox, David A. (2012), Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9.
1.2 External links • Larry Freeman. Fermat’s Last Theorem blog: Abel’s Lemmas on Irreducibility. September 4, 2008. • Weisstein, Eric W., “Abel’s Irreducibility Theorem”, MathWorld.
1
Chapter 2
Abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. A Galois extension is called solvable if its Galois group is solvable, i.e. if it is constructed from a series of abelian groups corresponding to intermediate extensions. Every finite extension of a finite field is a cyclic extension. The development of class field theory has provided detailed information about abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields. There are two slightly different concepts of cyclotomic extensions: these can mean either extensions formed by adjoining roots of unity, or subextensions of such extensions. The cyclotomic fields are examples. Any cyclotomic extension (for either definition) is abelian. If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting so-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th roots of elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product. The Kummer theory gives a complete description of the abelian extension case, and the Kronecker–Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity. There is an important analogy with the fundamental group in topology, which classifies all covering spaces of a space: abelian covers are classified by its abelianisation which relates directly to the first homology group.
2.1 References • Kuz'min, L.V. (2001), “cyclotomic extension”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
2
Chapter 3
Abhyankar’s inequality Abhyankar’s inequality is an inequality involving extensions of valued fields in algebra, introduced by Abhyankar (1956). If K/k is an extension of valued fields, then Abhyankar’s inequality states that the transcendence degree of K/k is at least the transcendence degree of the residue field extension plus the Q-rank of the quotient of the valuation groups.
3.1 References • Abhyankar, Shreeram (1956), “On the valuations centered in a local domain”, American Journal of Mathematics 78: 321–348, ISSN 0002-9327, JSTOR 2372519, MR 0082477
3
Chapter 4
Abhyankar’s lemma Abhyankar’s lemma is not directly related to Abhyankar’s conjecture. In mathematics, Abhyankar’s lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar’s lemma states that if A, B, C are local fields such that A and B are finite extensions of C, with ramification indices a and b, and B is tamely ramified over C and b divides a, then the compositum AB is an unramified extension of A.
4.1 References • Cornell, Gary (1982), “On the Construction of Relative Genus Fields”, Transactions of the American Mathematical Society 271 (2): 501–511, JSTOR 1998895. Theorem 3, page 504. • Gold, Robert; Madan, M. L. (1978), “Some applications of Abhyankar’s lemma”, Mathematische Nachrichten 82: 115–119, doi:10.1002/mana.19780820112. • Grothendieck, A. (1971), Revêtements étales et groupe fondamental (SGA 1, Séminaire de Géométrie Algébriques du Bois-Marie 1960/61), Lecture Notes in Mathematics 224, Springer-Verlag, arXiv:math.AG/0206203, p. 279. • Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3rd ed.), Berlin: Springer-Verlag, p. 229, ISBN 3-540-21902-1, Zbl 1159.11039.
4
Chapter 5
Abstract analytic number theory Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. The classical prime number theorem serves as a prototypical example, and the emphasis is on abstract asymptotic distribution results. The theory was invented and developed by mathematicians such as John Knopfmacher and Arne Beurling in the twentieth century.
5.1 Arithmetic semigroups The fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoid G satisfying the following properties: • There exists a countable subset (finite or countably infinite) P of G, such that every element a ≠ 1 in G has a unique factorisation of the form
αr 1 α2 a = pα 1 p2 · · · pr
where the pi are distinct elements of P, the αi are positive integers, r may depend on a, and two factorisations are considered the same if they differ only by the order of the factors indicated. The elements of P are called the primes of G. • There exists a real-valued norm mapping | | on G such that 1. |1| = 1 2. |p| > 1 for all p ∈ P 3. |ab| = |a||b| for all a, b ∈ G 4. The total number NG (x) of elements a ∈ G of norm |a| ≤ x is finite, for each real x > 0 .
5.1.1
Additive number systems
“Additive number system” redirects here. For food additive numbering, see E number. An additive number system is an arithmetic semigroup in which the underlying monoid G is free abelian. The norm function may be written additively.[1] If the norm is integer-valued, we associate counting functions a(n) and p(n) with G where p counts the number of elements of P of norm n, and a counts the number of elements of G of norm n. We let A(x) and P(x) be the corresponding formal power series. We have the fundamental identity[2] 5
6
CHAPTER 5. ABSTRACT ANALYTIC NUMBER THEORY
A(x) =
∑
a(n)xn =
n
∏ (1 − xn )−p(n) n
which formally encodes the unique expression of each element of G as a product of elements of P. The radius of convergence of G is the radius of convergence of the power series A(x).[3] The fundamental identity has the alternative form[4]
∑ P (xm ) . A(x) = exp m m≥1
5.2 Examples • The prototypical example of an arithmetic semigroup is the multiplicative semigroup of positive integers G = Z+ = {1, 2, 3, ...}, with subset of rational primes P = {2, 3, 5, ...}. Here, the norm of an integer is simply |n| = n , so that NG (x) = ⌊x⌋ , the greatest integer not exceeding x. • If K is an algebraic number field, i.e. a finite extension of the field of rational numbers Q, then the set G of all nonzero ideals in the ring of integers OK of K forms an arithmetic semigroup with identity element OK and the norm of an ideal I is given by the cardinality of the quotient ring OK/I. In this case, the appropriate generalisation of the prime number theorem is the Landau prime ideal theorem, which describes the asymptotic distribution of the ideals in OK. • Various arithmetical categories which satisfy a theorem of Krull-Schmidt type can be considered. In all these cases, the elements of G are isomorphism classes in an appropriate category, and P consists of all isomorphism classes of indecomposable objects, i.e. objects which cannot be decomposed as a direct product of nonzero objects. Some typical examples are the following. • The category of all finite abelian groups under the usual direct product operation and norm mapping |A| = card(A) . The indecomposable objects are the cyclic groups of prime power order. • The category of all compact simply-connected globally symmetric Riemannian manifolds under the Riemannian product of manifolds and norm mapping |M | = cdim M , where c > 1 is fixed, and dim M denotes the manifold dimension of M. The indecomposable objects are the compact simply-connected irreducible symmetric spaces. • The category of all pseudometrisable finite topological spaces under the topological sum and norm mapping |X| = 2card(X) . The indecomposable objects are the connected spaces.
5.3 Methods and techniques The use of arithmetic functions and zeta functions is extensive. The idea is to extend the various arguments and techniques of arithmetic functions and zeta functions in classical analytic number theory to the context of an arbitrary arithmetic semigroup which may satisfy one or more additional axioms. Such a typical axiom is the following, usually called “Axiom A” in the literature: • Axiom A. There exist positive constants A and δ , and a constant ν with 0 ≤ ν < δ , such that NG (x) = Axδ + O(xν ) as x → ∞. [5] For any arithmetic semigroup which satisfies Axiom A, we have the following abstract prime number theorem:[6]
πG (x) ∼
xδ as x → ∞ δ log x
where πG(x) = total number of elements p in P of norm |p| ≤ x.
5.4. SEE ALSO
5.3.1
7
Arithmetical formation
The notion of arithmetical formation provides a generalisation of the ideal class group in algebraic number theory and allows for abstract asymptotic distribution results under constraints. In the case of number fields, for example, this is Chebotarev’s density theorem. An arithmetical formation is an arithmetic semigroup G with an equivalence relation ≡ such that the quotient G/≡ is a finite abelian group A. This quotient is the class group of the formation and the equivalence classes are generalised arithmetic progressions or generalised ideal classes. If χ is a character of A then we can define a Dirichlet series ∑
χ([g])|g|−s
g∈G
which provides a notion of zeta function for arithmetical semigroup.[7]
5.4 See also • Axiom A, a property of dynamical systems • Beurling zeta function
5.5 References [1] Burris (2001) p.20 [2] Burris (2001) p.26 [3] Burris (2001) p.31 [4] Burris (2001) p.34 [5] Knopfmacher (1990) p.75 [6] Knopfmacher (1990) p.154 [7] Knopfmacher (1990) pp.250–264
• Burris, Stanley N. (2001). Number theoretic density and logical limit laws. Mathematical Surveys and Monographs 86. Providence, RI: American Mathematical Society. ISBN 0-8218-2666-2. Zbl 0995.11001. • Knopfmacher, John (1990) [1975]. Abstract Analytic Number Theory (2nd ed.). New York, NY: Dover Publishing. ISBN 0-486-66344-2. Zbl 0743.11002. • Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cambridge studies in advanced mathematics 97. p. 278. ISBN 0-521-84903-9. Zbl 1142.11001.
Chapter 6
Additive polynomial In mathematics, the additive polynomials are an important topic in classical algebraic number theory.
6.1 Definition Let k be a field of characteristic p, with p a prime number. A polynomial P(x) with coefficients in k is called an additive polynomial, or a Frobenius polynomial, if
P (a + b) = P (a) + P (b) as polynomials in a and b. It is equivalent to assume that this equality holds for all a and b in some infinite field containing k, such as its algebraic closure. Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition that P(a + b) = P(a) + P(b) for all a and b in the field. For infinite fields the conditions are equivalent, but for finite fields they are not, and the weaker condition is the “wrong” one and does not behave well. For example, over a field of order q any multiple P of xq − x will satisfy P(a + b) = P(a) + P(b) for all a and b in the field, but will usually not be (absolutely) additive.
6.2 Examples The polynomial xp is additive. Indeed, for any a and b in the algebraic closure of k one has by the binomial theorem p ( ) ∑ p
p
(a + b) =
n=0
n
an bp−n .
Since p is prime, for all n = 1, ..., p−1 the binomial coefficient (np ) is divisible by p, which implies that
(a + b)p ≡ ap + bp
mod p
as polynomials in a and b. Similarly all the polynomials of the form
n
τpn (x) = xp
are additive, where n is a non-negative integer. 8
6.3. THE RING OF ADDITIVE POLYNOMIALS
9
6.3 The ring of additive polynomials It is quite easy to prove that any linear combination of polynomials τpn (x) with coefficients in k is also an additive polynomial. An interesting question is whether there are other additive polynomials except these linear combinations. The answer is that these are the only ones. One can check that if P(x) and M(x) are additive polynomials, then so are P(x) + M(x) and P(M(x)). These imply that the additive polynomials form a ring under polynomial addition and composition. This ring is denoted
k{τp }. This ring is not commutative unless k equals the field Fp =Z/pZ (see modular arithmetic). Indeed, consider the additive polynomials ax and xp for a coefficient a in k. For them to commute under composition, we must have
(ax)p = axp , or ap − a = 0. This is false for a not a root of this equation, that is, for a outside Fp .
6.4 The fundamental theorem of additive polynomials Let P(x) be a polynomial with coefficients in k, and {w1 ,...,wm }⊂k be the set of its roots. Assuming that the roots of P(x) are distinct (that is, P(x) is separable), then P(x) is additive if and only if the set {w1 ,...,wm } forms a group with the field addition.
6.5 See also • Drinfeld module • Additive map
6.6 References • David Goss, Basic Structures of Function Field Arithmetic, 1996, Springer, Berlin. ISBN 3-540-61087-1.
6.7 External links • Weisstein, Eric W., “Additive Polynomial”, MathWorld.
Chapter 7
Adele ring In algebraic number theory and topological algebra, the adele ring[1] (other names are the adelic ring, the ring of adeles) is a self-dual topological ring built on the field of rational numbers (or, more generally, any algebraic number field). It involves in a symmetric way all the completions of the field. The adele ring was introduced by Claude Chevalley for the purposes of simplifying and clarifying class field theory. It has also found applications outside that area. The adele ring and its relation to the number field are among the most fundamental objects in number theory. The quotient of its multiplicative group by the multiplicative group of the algebraic number field is the central object in class field theory. It is a central principle of Diophantine geometry to study solutions of polynomials equations in number fields by looking at their solutions in the larger complete adele ring, where it is generally easier to detect solutions, and then deciding which of them come from the number field. The word “adele” is short for “additive idele"[2] and it was invented by André Weil. The previous name was the valuation vectors. The ring of adeles was historically preceded by the ring of repartitions, a construction which avoids completions, and is today sometimes referred to as pre-adele.
7.1 Definitions b , is the inverse limit of the rings Z/nZ : The profinite completion of the integers, Z b = lim Z/nZ. Z ←− By the Chinese remainder theorem it is isomorphic to the product of all the rings of p-adic integers: b= Z
∏
Zp .
p
The ring of integral adeles AZ is the product b AZ = R × Z. The ring of (rational) adeles AQ is the tensor product
A Q = Q ⊗Z A Z (topologized so that AZ is an open subring). More generally the ring of adeles AF of any algebraic number field F is the tensor product 10
7.2. PROPERTIES
11
AF = F ⊗Z AZ (topologized as the product of deg(F ) copies of AQ). The ring of (rational) adeles can also be defined as the restricted product
AQ = R ×
∏′
Qp
p
of all the p-adic completions Qp and the real numbers (or in other words as the restricted product of all completions of the rationals). In this case the restricted product means that for an adele (a∞, a2 , a3 , a5 , …) all but a finite number of the ap are p-adic integers.[2] The adeles of a function field over a finite field can be defined in a similar way, as the restricted product of all completions.
7.2 Properties The additive group of the adele ring is a locally compact complete group with respect to its most natural topology. This group is self-dual in the sense that it is topologically isomorphic to its group of characters. The adelic ring contains the number or function field as a discrete co-compact subgroup. Similarly, the multiplicative group of adeles, called the group of ideles, is a locally compact group with respect to its topology defined below.
7.3 Idele group The group of invertible elements of the adele ring is the idele group.[2][3] It is not given the subset topology, as the operation of inversion is not continuous in this topology. Instead the ideles are identified with the closed subset of all pairs (x,y) of A×A with xy=1, with the subset topology. The idele group may be realised as the restricted product of the unit groups of the local fields with respect to the subgroup of local integral units.[4] The ideles form a locally compact topological group.[5] The principal ideles are given by the diagonal embedding of the invertible elements of the number field or field of functions and the quotient of the idele group by principal ideles is the idele class group.[6] This is a key object of class field theory which describes abelian extensions of the field. The product of the local reciprocity maps in local class field theory gives a homomorphism from the idele group to the Galois group of the maximal abelian extension of the number or function field. The Artin reciprocity law, which is a high level generalization of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus we obtain the global reciprocity map from the idele class group to the abelian part of the absolute Galois group of the field.[7]
7.4 Applications The self-duality of the adeles of the function field of a curve over a finite field easily implies the Riemann–Roch theorem for the curve and the duality theory for the curve. As a locally compact abelian group, the adeles have a nontrivial translation invariant measure. Similarly, the group of ideles has a nontrivial translation invariant measure using which one defines a zeta integral. The latter was explicitly introduced in papers of Kenkichi Iwasawa and John Tate. The zeta integral allows one to study several key properties of the zeta function of the number field or function field in a beautiful concise way, reducing its functional equation of meromorphic continuation to a simple application of harmonic analysis and self-duality of the adeles, see Tate’s thesis.[8] The ring A combined with the theory of algebraic groups leads to adelic algebraic groups. For the function field of a smooth curve over a finite field the quotient of the multiplicative group (i.e. GL(1)) of its adele ring by the multiplicative group of the function field of the curve and units of integral adeles, i.e. those with integral local
12
CHAPTER 7. ADELE RING
components, is isomorphic to the group of isomorphisms of linear bundles on the curve, and thus carries a geometric information. Replacing GL(1) by GL(n), the corresponding quotient is isomorphic to the set of isomorphism classes of n vector bundles on the curve, as was already observed by André Weil. Another key object of number theory is automorphic representations of adelic GL(n) which are constituents of the space of square integrable complex valued functions on the quotient by GL(n) of the field. They play the central role in the Langlands correspondence which studies finite-dimensional representations of the Galois group of the field and which is one of noncommutative extensions of class field theory. Another development of the theory is related to the Tamagawa number for an adelic linear algebraic group. This is a volume measure relating G(Q) with G(A), saying how G(Q), which is a discrete group in G(A), lies in the latter. A conjecture of André Weil was that the Tamagawa number was always 1 for a simply connected G. This arose out of Weil’s modern treatment of results in the theory of quadratic forms; the proof was case-by-case and took decades, the final steps were taken by Robert Kottwitz in 1988 and V. I. Chernousov in 1989. The influence of the Tamagawa number idea was felt in the theory of arithmetic of abelian varieties through its use in the statement of the Birch and Swinnerton-Dyer conjecture, and through the Tamagawa number conjecture developed by Spencer Bloch, Kazuya Kato and many other mathematicians.
7.5 See also • Schwartz–Bruhat function
7.6 Notes [1] Also spelled: adèle ring /əˈdɛl rɪŋ/. [2] Neukirch (1999) p. 357. [3] William Stein, “Algebraic Number Theory”, May 4, 2004, p. 5. [4] Neukirch (1999) pp. 357–358. [5] Neukirch (1999) p. 361. [6] Neukirch (1999) pp. 358–359. [7] Cohen, Henri; Stevenhagen, Peter (2008). “Computational class field theory”. In Buhler, J.P.; P., Stevenhagen. Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. MSRI Publications 44. Cambridge University Press. pp. 497–534. ISBN 978-0-521-20833-8. Zbl 1177.11095. [8] Neukirch (1999) p. 503
7.7 References Almost any book on modern algebraic number theory, such as: • Fröhlich, A.; Cassels, J. W. (1967), Algebraic number theory, London and New York: Academic Press, ISBN 978-0-12-163251-9, Zbl 0153.07403 • Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics 110 (2nd ed.), New York: Springer-Verlag, ISBN 978-0-387-94225-4, MR 1282723, Zbl 0811.11001 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
Chapter 8
Adelic algebraic group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear algebraic group. In the case of G an abelian variety it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms. In case G is a linear algebraic group, it is an affine algebraic variety in affine N-space. The topology on the adelic algebraic group G(A) is taken to be the subspace topology in AN , the Cartesian product of N copies of the adele ring.
8.1 Ideles An important example, the idele group I(K), is the case of G = GL1 . Here the set of ideles (also idèles /ɪˈdɛlz/) consists of the invertible adeles; but the topology on the idele group is not their topology as a subset of the adeles. Instead, considering that GL1 lies in two-dimensional affine space as the 'hyperbola' defined parametrically by {(t, t −1 )}, the topology correctly assigned to the idele group is that induced by inclusion in A2 ; composing with a projection, it follows that the ideles carry a finer topology than the subspace topology from A. Inside AN , the product K N lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. In the case of the idele group, the quotient group I(K)/K × is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a compact group; the proof of this is essentially equivalent to the finiteness of the class number. The study of the Galois cohomology of idele class groups is a central matter in class field theory. Characters of the idele class group, now usually called Hecke characters, give rise to the most basic class of L-functions.
8.2 Tamagawa numbers See also: Weil conjecture on Tamagawa numbers For more general G, the Tamagawa number is defined (or indirectly computed) as the measure of G(A)/G(K). 13
14
CHAPTER 8. ADELIC ALGEBRAIC GROUP
Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on G, defined over K, the measure involved was well-defined: while ω could be replaced by cω with c a non-zero element of K, the product formula for valuations in K is reflected by the independence from c of the measure of the quotient, for the product measure constructed from ω on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory.
8.3 History of the terminology Historically the idèles were introduced by Chevalley (1936) under the name "élément idéal”, which is “ideal element” in French, which Chevalley (1940) then abbreviated to “idèle” following a suggestion of Hasse. (In these papers he also gave the ideles a non-Hausdorff topology.) This was to formulate class field theory for infinite extensions in terms of topological groups. Weil (1938) defined (but did not name) the ring of adeles in the function field case and pointed out that Chevalley’s group of Idealelemente was the group of invertible elements of this ring. Tate (1950) defined the ring of adeles as a restricted direct product, though he called its elements “valuation vectors” rather than adeles. Chevalley (1951) defined the ring of adeles in the function field case, under the name “repartitions”. The term adèle (short for additive idèles, and also a French woman’s name) was in use shortly afterwards (Jaffard 1953) and may have been introduced by André Weil. The general construction of adelic algebraic groups by Ono (1957) followed the algebraic group theory founded by Armand Borel and Harish-Chandra.
8.4 References • Chevalley, Claude (1936), “Généralisation de la théorie du corps de classes pour les extensions infinies.”, Journal de Mathématiques Pures et Appliquées (in French) 15: 359–371, JFM 62.1153.02 • Chevalley, Claude (1940), “La théorie du corps de classes”, Annals of Mathematics. Second Series 41: 394– 418, ISSN 0003-486X, JSTOR 1969013, MR 0002357 • Chevalley, Claude (1951), Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, Providence, R.I.: American Mathematical Society, MR 0042164 • Jaffard, Paul (1953), Anneaux d'adèles (d'après Iwasawa), Séminaire Bourbaki, Secrétariat mathématique, Paris, MR 0157859 • Ono, Takashi (1957), “Sur une propriété arithmétique des groupes algébriques commutatifs”, Bulletin de la Société Mathématique de France 85: 307–323, ISSN 0037-9484, MR 0094362 • Tate, John T. (1950), “Fourier analysis in number fields, and Hecke’s zeta-functions”, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-09502734-2-6, MR 0217026 • Weil, André (1938), “Zur algebraischen Theorie der algebraischen Funktionen.”, Journal für Reine und Angewandte Mathematik (in German) 179: 129–133, doi:10.1515/crll.1938.179.129, ISSN 0075-4102
8.5 External links • Rapinchuk, A.S. (2001), “Tamagawa number”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 9
Adjunction (field theory) In abstract algebra, adjunction is a construction in field theory, where for a given field extension E/F, subextensions between E and F are constructed.
9.1 Definition Let E be a field extension of a field F. Given a set of elements A in the larger field E we denote by F(A) the smallest subextension which contains the elements of A. We say F(A) is constructed by adjunction of the elements A to F or generated by A. If A is finite we say F(A) is finitely generated and if A consists of a single element we say F(A) is a simple extension. The primitive element theorem states a finite separable extension is simple. In a sense, a finitely generated extension is a transcendental generalization of a finite extension since, if the generators in A are all algebraic, then F(A) is a finite extension of F. Because of this, most examples come from algebraic geometry. A subextension of a finitely generated field extension is also a finitely generated extension.[1]
9.2 Notes F(A) consists of all those elements of E that can be constructed using a finite number of field operations +, -, *, / applied to elements from F and A. For this reason F(A) is sometimes called the field of rational expressions in F and A.
9.3 Examples • Given a field extension E/F then F(Ø) = F and F(E) = E. • The complex numbers are constructed by adjunction of the imaginary unit to the real numbers, that is C=R(i).
9.4 Properties Given a field extension E/F and a subset A of E, let T be the family of all finite subsets of A. Then
F (A) =
∪
F (T )
T ∈T
In other words the adjunction of any set can be reduced to a union of adjunctions of finite sets. 15
16
CHAPTER 9. ADJUNCTION (FIELD THEORY)
Given a field extension E/F and two subsets N, M of E then K(M ∪ N) = (K(M))(N) = (K(N))(M). This shows that any adjunction of a finite set can be reduced to a successive adjunction of single elements.
9.5 References [1] Kolchin, E. R. (1973), Differential Algebra & Algebraic Groups, Pure and Applied Mathematics 54, Academic Press, p. 112, ISBN 9780080873695.
Chapter 10
Albert–Brauer–Hasse–Noether theorem In algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field K which splits over every completion Kv is a matrix algebra over K. The theorem is an example of a local-global principle in algebraic number theory and leads to a complete description of finite-dimensional division algebras over algebraic number fields in terms of their local invariants. It was proved independently by Helmut Hasse, Richard Brauer, and Emmy Noether and by Abraham Adrian Albert.
10.1 Statement of the theorem Let A be a central simple algebra of rank d over an algebraic number field K. Suppose that for any valuation v, A splits over the corresponding local field Kv:
A ⊗K Kv ≃ Md (Kv ). Then A is isomorphic to the matrix algebra Md(K).
10.2 Applications Using the theory of Brauer group, one shows that two central simple algebras A and B over an algebraic number field K are isomorphic over K if and only if their completions Av and Bv are isomorphic over the completion Kv for every v. Together with the Grunwald–Wang theorem, the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is cyclic, i.e. can be obtained by an explicit construction from a cyclic field extension L/K .
10.3 See also • Class field theory • Hasse norm theorem
10.4 References • Albert, A.A.; Hasse, H. (1932), “A determination of all normal division algebras over an algebraic number field”, Trans. Amer. Math. Soc. 34 (3): 722–726, doi:10.1090/s0002-9947-1932-1501659-x, Zbl 0005.05003 17
18
CHAPTER 10. ALBERT–BRAUER–HASSE–NOETHER THEOREM • Hasse, H.; Brauer, R.; Noether, E. (1931), “Beweis eines Hauptsatzes in der Theorie der Algebren”, Journal für Mathematik 167: 399–404 • Fenster, D.D.; Schwermer, J. (2005), “Delicate collaboration: Adrian Albert and Helmut Hasse and the Principal Theorem in Division Algebras” (PDF), Archive for history of exact sciences 59 (4): 349–379, doi:10.1007/s00407004-0093-6, retrieved 2009-07-05 • Pierce, Richard (1982), Associative algebras, Graduate Texts in Mathematics 88, New York-Berlin: SpringerVerlag, ISBN 0-387-90693-2, Zbl 0497.16001 • Reiner, I. (2003), Maximal Orders, London Mathematical Society Monographs. New Series 28, Oxford University Press, p. 276, ISBN 0-19-852673-3, Zbl 1024.16008 • Roquette, Peter (2005), “The Brauer–Hasse–Noether theorem in historical perspective” (PDF), Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften 15, MR 2222818, Zbl 1060.01009, CiteSeerX: 10.1.1.72.4101, retrieved 2009-07-05 Revised version — Roquette, Peter (2013), Contributions to the history of number theory in the 20th century, Heritage of European Mathematics, Zürich: European Mathematical Society, pp. 1–76, ISBN 978-3-03719-113-2, Zbl 1276.11001 • Albert, Nancy E. (2005), “A Cubed & His Algebra, iUniverse, isbn-13: 978-0-595-32817-8
10.5 Notes
Chapter 11
Algebraic closure For other uses, see Closure (disambiguation). In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn’s lemma, it can be shown that every field has an algebraic closure,[1][2][3] and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K. The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K. The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.[3]
11.1 Examples • The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers. • The algebraic closure of the field of rational numbers is the field of algebraic numbers. • There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q(π). • For a finite field of prime power order q, the algebraic closure is a countably infinite field that contains a copy of the field of order qn for each positive integer n (and is in fact the union of these copies).[4]
11.2 Existence of an algebraic closure and splitting fields Let S = {fλ |λ ∈ Λ} be the set of all monic irreducible polynomials in K[x]. For each fλ ∈ S , introduce new variables uλ,1 , . . . , uλ,d where d = degree(fλ ) . Let R be the polynomial ring over K generated by uλ,i for all λ ∈ Λ and all i ≤ degree(fλ ) . Write
fλ −
d ∏
(x − uλ,i ) =
i=1
d−1 ∑
rλ,j · xj ∈ R[x]
j=0
19
20
CHAPTER 11. ALGEBRAIC CLOSURE
with rλ,j ∈ R . Let I be the ideal in R generated by the rλ,j . By Zorn’s lemma, there exists a maximal ideal M in R that contains I. Now R/M is an algebraic closure of K; every fλ splits as the product of the x − (uλ,i + M ) . The same proof also shows that for any subset S of K[x], there exists a splitting field of S over K.
11.3 Separable closure An algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separable extensions of K within Kalg . This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Ksep , of degree > 1. Saying this another way, K is contained in a separably-closed algebraic extension field. It is essentially unique (up to isomorphism).[5] The separable closure is the full algebraic closure if and √ only if K is a perfect field. For example, if K is a field of characteristic p and if X is transcendental over K, K(X)( p X) ⊃ K(X) is a non-separable algebraic field extension. In general, the absolute Galois group of K is the Galois group of Ksep over K.[6]
11.4 See also • Algebraically closed field • Algebraic extension • Puiseux expansion
11.5 References [1] McCarthy (1991) p.21 [2] M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. Addison-Wesley publishing Company. pp. 11-12. [3] Kaplansky (1972) pp.74-76 [4] Brawley, Joel V.; Schnibben, George E. (1989), “2.2 The Algebraic Closure of a Finite Field”, Infinite Algebraic Extensions of Finite Fields, Contemporary Mathematics 95, American Mathematical Society, pp. 22–23, ISBN 978-0-8218-5428-0, Zbl 0674.12009. [5] McCarthy (1991) p.22 [6] Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.
• Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University of Chicago Press. ISBN 0-226-42451-0. Zbl 1001.16500. • McCarthy, Paul J. (1991). Algebraic extensions of fields (Corrected reprint of the 2nd ed.). New York: Dover Publications. Zbl 0768.12001.
Chapter 12
Algebraic extension In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental. For example, the field extension R/Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C/R and Q(√2)/Q are algebraic, where C is the field of complex numbers. All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic.[1] The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers. If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. The converse is true as well, if K[a] is a field, then a is algebraic over K. In the special case where K = Q is the field of rational numbers, Q[a] is an example of an algebraic number field. A field with no nontrivial algebraic extensions is called algebraically closed. An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice. An extension L/K is algebraic if and only if every sub K-algebra of L is a field.
12.1 Properties The class of algebraic extensions forms a distinguished class of field extensions, that is, the following three properties hold:[2] 1. If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K. 2. If E and F are algebraic extensions of K in a common overfield C, then the compositum EF is an algebraic extension of K. 3. If E is an algebraic extension of F and E>K>F then E is an algebraic extension of K. These finitary results can be generalized using transfinite induction: 1. The union of any chain of algebraic extensions over a base field is itself an algebraic extension over the same base field. This fact, together with Zorn’s lemma (applied to an appropriately chosen poset), establishes the existence of algebraic closures. 21
22
CHAPTER 12. ALGEBRAIC EXTENSION
12.2 Generalizations Main article: Substructure Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p(x) is true and the set { } y ∈ N p(y) is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The Galois group of N over M can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case.
12.3 See also • Integral element • Lüroth’s theorem • Galois extension • Separable extension • Normal extension
12.4 Notes [1] See also Hazewinkel et al. (2004), p. 3. [2] Lang (2002) p.228
12.5 References • Hazewinkel, Michiel; Gubareni, Nadiya; Gubareni, Nadezhda Mikhaĭlovna; Kirichenko, Vladimir V. (2004), Algebras, rings and modules 1, Springer, ISBN 1-4020-2690-0 • Lang, Serge (1993), “V.1:Algebraic Extensions”, Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., pp. 223ff, ISBN 978-0-201-55540-0, Zbl 0848.13001 • McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], Algebraic extensions of fields, New York: Dover Publications, ISBN 0-486-66651-4, Zbl 0768.12001 • Roman, Steven (1995), Field Theory, GTM 158, Springer-Verlag, ISBN 9780387944081 • Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 9780130878687
Chapter 13
Algebraic function field In mathematics, an (algebraic) function field of n variables over the field k is a finitely generated field extension K/k which has transcendence degree n over k.[1] Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field k(x1 ,...,xn) of rational functions in n variables over k.
13.1 Example As an example, in the polynomial ring k[X,Y] consider the ideal generated by the irreducible polynomial Y 2 −X3 and form the field of fractions√of the quotient ring k[X,Y]/(Y 2 −X3 ). This is a√function field of one variable over k; it can 3 also be written as k(X)( X 3 ) (with degree 2 over k(X) ) or as k(Y )( Y 2 ) (with degree 3 over k(Y ) ). We see that the degree of an algebraic function field is not a well-defined notion.
13.2 Category structure The algebraic function fields over k form a category; the morphisms from function field K to L are the ring homomorphisms f : K→L with f(a)=a for all a∈k. All these morphisms are injective. If K is a function field over k of n variables, and L is a function field in m variables, and n>m, then there are no morphisms from K to L.
13.3 Function fields arising from varieties, curves and Riemann surfaces The function field of an algebraic variety of dimension n over k is an algebraic function field of n variables over k. Two varieties are birationally equivalent if and only if their function fields are isomorphic. (But note that non-isomorphic varieties may have the same function field!) Assigning to each variety its function field yields a duality (contravariant equivalence) between the category of varieties over k (with dominant rational maps as morphisms) and the category of algebraic function fields over k. (Note that the varieties considered here are to be taken in the scheme sense; they need not have any k-rational points, like the curve X2 +Y 2 +1=0 over R.) The case n=1 (irreducible algebraic curves in the scheme sense) is especially important, since every function field of one variable over k arises as the function field of a uniquely defined regular (i.e. non-singular) projective irreducible algebraic curve over k. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with dominant regular maps as morphisms) and the category of function fields of one variable over k. The field M(X) of meromorphic functions defined on a connected Riemann surface X is a function field of one variable over the complex numbers C. In fact, M yields a duality (contravariant equivalence) between the category of compact connected Riemann surfaces (with non-constant holomorphic maps as morphisms) and function fields of one variable over C. A similar correspondence exists between compact connected Klein surfaces and function fields in one variable over R. 23
24
CHAPTER 13. ALGEBRAIC FUNCTION FIELD
13.4 Number fields and finite fields The function field analogy states that almost all theorems on number fields have a counterpart on function fields of one variable over a finite field, and these counterparts are frequently easier to prove. (For example, see Analogue for irreducible polynomials over a finite field.) In the context of this analogy, both number fields and function fields over finite fields are usually called "global fields". The study of function fields over a finite field has applications in cryptography and error correcting codes. For example, the function field of an elliptic curve over a finite field (an important mathematical tool for public key cryptography) is an algebraic function field. Function fields over the field of rational numbers play also an important role in solving inverse Galois problems.
13.5 Field of constants Given any algebraic function field K over k, we can consider the set of elements of K which are algebraic over k. These elements form a field, known as the field of constants of the algebraic function field. For instance, C(x) is a function field of one variable over R; its field of constants is C.
13.6 Valuations and places Key tools to study algebraic function fields are absolute values, valuations, places and their completions. Given an algebraic function field K/k of one variable, we define the notion of a valuation ring of K/k: this is a subring O of K that contains k and is different from k and K, and such that for any x in K we have x∈O or x −1 ∈O. Each such valuation ring is a discrete valuation ring and its maximal ideal is called a place of K/k. A discrete valuation of K/k is a surjective function v : K→Zu{∞} such that v(x)=∞ iff x=0, v(xy)=v(x)+v(y) and v(x+y)≥min(v(x),v(y)) for all x,y∈K, and v(a)=0 for all a∈k\{0}. There are natural bijective correspondences between the set of valuation rings of K/k, the set of places of K/k, and the set of discrete valuations of K/k. These sets can be given a natural topological structure: the Zariski–Riemann space of K/k. In case k is algebraically closed, the Zariski-Riemann space of K/k is a smooth curve over k and K is the function field of this curve.
13.7 See also • function field of an algebraic variety • function field (scheme theory) • algebraic function
13.8 References [1] Gabriel Daniel and Villa Salvador (2007). Topics in the Theory of Algebraic Function Fields. Springer.
Chapter 14
Algebraic number field In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
14.1 Definition 14.1.1
Prerequisites
Main articles: Field and Vector space The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted Q, together with its usual operations of addition etc. Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples) (x1 , x2 , ...) whose entries are elements of a fixed field, such as the field Q. Any two such sequences can be added by adding the entries one per one. Furthermore, any sequence can be multiplied by a single element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be “infinite-dimensional”, that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector space consists of finite sequences (x1 , x2 , ..., xn), the vector space is said to be of finite dimension, n.
14.1.2
Definition
An algebraic number field (or simply number field) is a finite degree field extension of the field of rational numbers. Here its dimension as a vector space over Q is simply called its degree. 25
26
CHAPTER 14. ALGEBRAIC NUMBER FIELD
14.2 Examples • The smallest and most basic number field is the field Q of rational numbers. Many properties of general number fields, such as unique factorization, are modelled after the properties of Q. • The Gaussian rationals, denoted Q(i) (read as "Q adjoined i"), form the first nontrivial example of a number field. Its elements are expressions of the form a+bi where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity i2 = −1. Explicitly, (a + bi) + (c + di) = (a + c) + (b + d)i, (a + bi) (c + di) = (ac − bd) + (ad + bc)i. Non-zero Gaussian rational numbers are invertible, which can be seen from the identity
( (a + bi)
) a b (a + bi)(a − bi) − 2 i = = 1. 2 2 2 a +b a +b a2 + b2
It follows that the Gaussian rationals form a number field which is two-dimensional as a vector space over Q. • More generally, for any square-free integer d, the quadratic field Q(√d) is a number field obtained by adjoining the square root of d to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of gaussian rational numbers, d = − 1. • Cyclotomic field Q(ζn), ζn = exp (2πi / n) is a number field obtained from Q by adjoining a primitive nth root of unity ζn. This field contains all complex nth roots of unity and its dimension over Q is equal to φ(n), where φ is the Euler totient function. • The real numbers, R, and the complex numbers, C, are fields which have infinite dimension as Q-vector spaces, hence, they are not number fields. This follows from the uncountability of R and C as sets, whereas every number field is necessarily countable. • The set Q2 of ordered pairs of rational numbers, with the entrywise addition and multiplication is a twodimensional commutative algebra over Q. However, it is not a field, since it has zero divisors: (1, 0) · (0, 1) = (1 · 0, 0 · 1) = (0, 0).
14.3. ALGEBRAICITY AND RING OF INTEGERS
27
14.3 Algebraicity and ring of integers Generally, in abstract algebra, a field extension F / E is algebraic if every element f of the bigger field F is the zero of a polynomial with coefficients e0 , ..., em in E: p(f) = emf m + em₋₁f m−1 + ... + e1 f + e0 = 0. It is a fact that every field extension of finite degree is algebraic (proof: for x in F simply consider 1, x, x2 , x3 , ..., we get a linear dependence, i.e. a polynomial that x is a root of!) because of the finite degree. In particular this applies to algebraic number fields, so any element f of an algebraic number field F can be written as a zero of a polynomial with rational coefficients. Therefore, elements of F are also referred to as algebraic numbers. Given a polynomial p such that p(f) = 0, it can be arranged such that the leading coefficient em is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rational coefficients. If, however, its coefficients are actually all integers, f is called an algebraic integer. Any (usual) integer z ∈ Z is an algebraic integer, as it is the zero of the linear monic polynomial: p(t) = t − z. It can be shown that any algebraic integer that is also a rational number must actually be an integer, whence the name “algebraic integer”. Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer, it follows that the algebraic integers in F form a ring denoted OF called the ring of integers of F. It is a subring of (that is, a ring contained in) F. A field contains no zero divisors and this property is inherited by any subring. Therefore, the ring of integers of F is an integral domain. The field F is the field of fractions of the integral domain OF. This way one can get back and forth between the algebraic number field F and its ring of integers OF. Rings of algebraic integers have three distinctive properties: firstly, OF is an integral domain that is integrally closed in its field of fractions F. Secondly, OF is a Noetherian ring. Finally, every nonzero prime ideal of OF is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.
14.3.1
Unique factorization and class number
For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals. However, unlike Z as the ring of integers of Q, the ring of integers of a proper extension of Q need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. This happens already for quadratic integers, for example in OQ₍√₋₅₎ = Z[√−5], the uniqueness of the factorization fails: 6 = 2 · 3 = (1 + √−5) · (1 − √−5). Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors do not just differ by a unit in OQ₍√₋₅₎. Euclidean domains are unique factorization domains; for example Z[i], the ring of Gaussian integers, and Z[ω], the ring of Eisenstein integers, where ω is a third root of unity (unequal to 1), have this property.[1]
14.3.2
ζ-functions, L-functions and class number formula
The failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of the so-called ideal class group. This group is always finite. The ring of integers OF possesses unique factorization if and only if it is a principal ring or, equivalently, if F has class number 1. Given a number field, the class number is often difficult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginary quadratic number fields (i.e., Q(√−d), d ≥ 1) with prescribed class number. The class number formula relates h to other fundamental invariants of F. It involves the Dedekind zeta function ζF(s), a function in a complex variable s, defined by
ζF (s) :=
∏ p
1 1 − N (p)−s
28
CHAPTER 14. ALGEBRAIC NUMBER FIELD
(The product is over all prime ideals of OF, N (p) denotes the norm of the prime ideal or, equivalently, the (finite) number of elements in the residue field OF /p . The infinite product converges only for Re(s) > 1, in general analytic continuation and the functional equation for the zeta-function are needed to define the function for all s). The Dedekind zeta-function generalizes the Riemann zeta-function in that ζQ(s) = ζ(s). The class number formula states that ζF(s) has a simple pole at s = 1 and at this point (its meromorphic continuation to the whole complex plane) the residue is given by 2r1 · (2π)r2 · h · Reg √ . w · |D| Here r1 and r2 classically denote the number of real embeddings and pairs of complex embeddings of F, respectively. Moreover, Reg is the regulator of F, w the number of roots of unity in F and D is the discriminant of F. Dirichlet L-functions L(χ, s) are a more refined variant of ζ(s). Both types of functions encode the arithmetic behavior of Q and F, respectively. For example, Dirichlet’s theorem asserts that in any arithmetic progression a, a + m, a + 2m, ... with coprime a and m, there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet L-function is nonzero at s = 1. Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture), of values of more general L-functions.[2]
14.4 Bases for number fields 14.4.1
Integral basis
An integral basis for a number field F of degree n is a set B = {b1 , …, bn} of n algebraic integers in F such that every element of the ring of integers OF of F can be written uniquely as a Z-linear combination of elements of B; that is, for any x in OF we have x = m1 b1 + … + mnbn, where the mi are (ordinary) integers. It is then also the case that any element of F can be written uniquely as m1 b1 + … + mnbn, where now the mi are rational numbers. The algebraic integers of F are then precisely those elements of F where the mi are all integers. Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.
14.4.2
Power basis
Let F be a number field of degree n. Among all possible bases of F (seen as a Q-vector space), there are particular ones known as power bases, that are bases of the form Bx = {1, x, x2 , ..., xn−1 } for some element x ∈ F. By the primitive element theorem, there exists such an x, called a primitive element. If x can be chosen in OF and such that Bx is a basis of OF as a free Z-module, then Bx is called a power integral basis, and the field F is called a monogenic field. An example of a number field that is not monogenic was first given by Dedekind. His example is the field obtained by adjoining a root of the polynomial x3 − x2 − 2x − 8.[3]
14.5. REGULAR REPRESENTATION, TRACE AND DETERMINANT
29
14.5 Regular representation, trace and determinant Using the multiplication in F, the elements of the field F may be represented by n-by-n matrices A = A(x)=(aij)₁ ≤ i, j ≤ n, by requiring
xei =
n ∑
aij ej ,
aij ∈ Q.
j=1
Here e1 , ..., en is a fixed basis for F, viewed as a Q-vector space. The rational numbers aij are uniquely determined by x and the choice of a basis since any element of F can be uniquely represented as a linear combination of the basis elements. This way of associating a matrix to any element of the field F is called the regular representation. The square matrix A represents the effect of multiplication by x in the given basis. It follows that if the element y of F is represented by a matrix B, then the product xy is represented by the matrix product BA. Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element x and not on the basis. In particular, the trace of the matrix A(x) is called the trace of the field element x and denoted Tr(x), and the determinant is called the norm of x and denoted N(x). By definition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is a linear function of x, as expressed by Tr(x + y) = Tr(x) + Tr(y), Tr(λx) = λ Tr(x), and the norm is a multiplicative homogeneous function of degree n: N(xy) = N(x) N(y), N(λx) = λn N(x). Here λ is a rational number, and x, y are any two elements of F. The trace form derives is a bilinear form defined by means of the trace, as Tr(x y). The integral trace form, an integervalued symmetric matrix is defined as tᵢ = Tr(bᵢb ), where b1 , ..., b is an integral basis for F. The discriminant of F is defined as det(t). It is an integer, and is an invariant property of the field F, not depending on the choice of integral basis. The matrix associated to an element x of F can also be used to give other, equivalent descriptions of algebraic integers. An element x of F is an algebraic integer if and only if the characteristic polynomial pA of the matrix A associated to x is a monic polynomial with integer coefficients. Suppose that the matrix A that represents an element x has integer entries in some basis e. By the Cayley–Hamilton theorem, pA(A) = 0, and it follows that pA(x) = 0, so that x is an algebraic integer. Conversely, if x is an element of F which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix A. In this case it can be proven that A is an integer matrix in a suitable basis of F. Note that the property of being an algebraic integer is defined in a way that is independent of a choice of a basis in F.
14.5.1
Example
Consider F = Q(x), where x satisfies x3 − 11x2 + x + 1 = 0. Then an integral basis is [1, x, 1/2(x2 + 1)], and the corresponding integral trace form is
3 11 61
11 119 653
61 653 . 3589
The “3” in the upper left hand corner of this matrix is the trace of the matrix of the map defined by the first basis element (1) in the regular representation of F on F. This basis element induces the identity map on the 3-dimensional vector space, F. The trace of the matrix of the identity map on a 3-dimensional vector space is 3. The determinant of this is 1304 = 23 ·163, the field discriminant; in comparison the root discriminant, or discriminant of the polynomial, is 5216 = 25 ·163.
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CHAPTER 14. ALGEBRAIC NUMBER FIELD
14.6 Places Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number.[4][5] This situation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field F into its various topological completions at once. A place of a number field F is an equivalence class of absolute values on F. Essentially, an absolute value is a notion to measure the size of elements f of F. Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity). In general, they fall into three regimes. Firstly (and mostly irrelevant), the trivial absolute value | |0 , which takes the value 1 on all non-zero f in F. The second and third classes are Archimedean places and non-Archimedean (or ultrametric) places. The completion of F with respect to a place is given in both cases by taking Cauchy sequences in F and dividing out null sequences, that is, sequences (xn)n ∈ N such that |xn| tends to zero when n tends to infinity. This can be shown to be a field again, the so-called completion of F at the given place. For F = Q, the following non-trivial norms occur (Ostrowski’s theorem): the (usual) absolute value, which gives rise to the complete topological field of the real numbers R. On the other hand, for any prime number p, the p-adic absolute values is defined by |q|p = p−n , where q = pn a/b and a and b are integers not divisible by p. In contrast to the usual absolute value, the p-adic norm gets smaller when q is multiplied by p, leading to quite different behavior of Qp vis-à-vis R.
14.6.1
Archimedean places
[6][7]
For some of the details take a look at,[8] Chapter 11 §C p. 108. Note in particular the standard notation r1 and r2 for the number of real and complex embeddings, respectively (see below). Calculating the archimedean places of F is done as follows: let x be a primitive element of F, with minimal polynomial (over Q) f. Over R, f will generally no longer be irreducible, but its irreducible (real) factors are either of degree one or two. Since there are no repeated roots, there are no repeated factors. The roots r of factors of degree one are necessarily real, and replacing x by r gives an embedding of F into R; the number of such embeddings is equal to the number of real roots of f. Restricting the standard absolute value on R to F gives an archimedean absolute value on F; such an absolute value is also referred to as a real place of F. On the other hand, the roots of factors of degree two are pairs of conjugate complex numbers, which allows for two conjugate embeddings into C. Either one of this pair of embeddings can be used to define an absolute value on F, which is the same for both embeddings since they are conjugate. This absolute value is called a complex place of F. If all roots of f above are real (respectively, complex) or, equivalently, any possible embedding F ⊂ C is actually forced to be inside R (resp. C), F is called totally real (resp. totally complex).
14.6.2
Nonarchimedean or ultrametric places
To find the nonarchimedean places, let again f and x be as above. In Qp, f splits in factors of various degrees, none of which are repeated, and the degrees of which add up to n, the degree of f. For each of these p-adically irreducible factors t, we may suppose that x satisfies t and obtain an embedding of F into an algebraic extension of finite degree over Q . Such a local field behaves in many ways like a number field, and the p-adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to Qp. By using this p-adic norm map Nt for the place t, we may define an absolute value corresponding to a given p-adically irreducible factor t of degree m by |θ|t = |Nt(θ)|p1/m . Such an absolute value is called an ultrametric, non-Archimedean or p-adic place of F. For any ultrametric place v we have that |x|v ≤ 1 for any x in OF, since the minimal polynomial for x has integer factors, and hence its p-adic factorization has factors in Zp. Consequently, the norm term (constant term) for each factor is a p-adic integer, and one of these is the integer used for defining the absolute value for v.
14.7. RAMIFICATION
14.6.3
31
Prime ideals in OF
For an ultrametric place v, the subset of OF defined by |x|v < 1 is an ideal P of OF. This relies on the ultrametricity of v: given x and y in P, then |x + y|v ≤ max (|x|v, |y|v) < 1. Actually, P is even a prime ideal. Conversely, given a prime ideal P of OF, a discrete valuation can be defined by setting vP(x) = n where n is the biggest integer such that x ∈ P n , the n-fold power of the ideal. This valuation can be turned into an ultrametric place. Under this correspondence, (equivalence classes) of ultrametric places of F correspond to prime ideals of OF. For F = Q, this gives back Ostrowski’s theorem: any prime ideal in Z (which is necessarily by a single prime number) corresponds to an non-archimedean place and vice versa. However, for more general number fields, the situation becomes more involved, as will be explained below. Yet another, equivalent way of describing ultrametric places is by means of localizations of OF. Given an ultrametric place v on a number field F, the corresponding localization is the subring T of F of all elements x such that | x |v ≤ 1. By the ultrametric property T is a ring. Moreover, it contains OF. For every element x of F, at least one of x or x−1 is contained in T. Actually, since F × /T × can be shown to be isomorphic to the integers, T is a discrete valuation ring, in particular a local ring. Actually, T is just the localization of OF at the prime ideal P. Conversely, P is the maximal ideal of T. Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.
14.7 Ramification
Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y.
Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps f: X → Y such that the preimages of all points y in Y consist only of finitely many points): the cardinality of the fibers f −1 (y) will generally have the same number of points, but it occurs that, in special points y, this number drops. For example, the map C → C, z ↦ zn has n points in each fiber over t, namely the n (complex) roots of t, except in t = 0, where the fiber consists of only one element, z = 0. One says that the map is “ramified” in zero. This is an example of a branched covering of Riemann surfaces. This intuition also serves to define ramification in algebraic number theory. Given a (necessarily finite) extension of number fields F / E, a prime ideal p of OE generates the ideal pOF of OF. This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by
32
CHAPTER 14. ALGEBRAIC NUMBER FIELD pOF = q1 e1 q2 e2 ... qmem
with uniquely determined prime ideals qi of OF and numbers (called ramification indices) ei. Whenever one ramification index is bigger than one, the prime p is said to ramify in F. The connection between this definition and the geometric situation is delivered by the map of spectra of rings Spec OF → Spec OE. In fact, unramified morphisms of schemes in algebraic geometry are a direct generalization of unramified extensions of number fields. Ramification is a purely local property, i.e., depends only on the completions around the primes p and qi. The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.
14.7.1
An example
The following example illustrates the notions introduced above. In order to compute the ramification index of Q(x), where f(x) = x3 − x − 1 = 0, at 23, it suffices to consider the field extension Q23 (x) / Q23 . Up to 529 = 232 (i.e., modulo 529) f can be factored as f(x) = (x + 181)(x2 − 181x − 38) = gh. Substituting x = y + 10 in the first factor g modulo 529 yields y + 191, so the valuation | y |g for y given by g is | −191 |23 = 1. On the other hand the same substitution in h yields y2 − 161y − 161 modulo 529. Since 161 = 7 × 23, |y|h = √∣161∣23 = 1 / √23. Since possible values for the absolute value of the place defined by the factor h are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two. The valuations of any element of F can be computed in this way using resultants. If, for example y = x2 − x − 1, using the resultant to eliminate x between this relationship and f = x3 − x − 1 = 0 gives y3 − 5y2 + 4y − 1 = 0. If instead we eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g and h (which are both 1 in this instance.)
14.7.2
Dedekind discriminant theorem
Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in Qp where p divides the discriminant. This is even true of the polynomial discriminant; however the converse is also true, that if a prime p divides the discriminant, then there is a p-place which ramifies. For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field Q(x) with x3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place which does. The other ramified place comes from the absolute value on the complex embedding of F.
14.8 Galois groups and Galois cohomology Generally in abstract algebra, field extensions F / E can be studied by examining the Galois group Gal(F / E), consisting of field automorphisms of F leaving E elementwise fixed. As an example, the Galois group Gal (Q(ζn) / Q) of the cyclotomic field extension of degree n (see above) is given by (Z/nZ)× , the group of invertible elements in Z/nZ. This is the first stepstone into Iwasawa theory. In order to include all possible extensions having certain properties, the Galois group concept is commonly applied to the (infinite) field extension F / F of the algebraic closure, leading to the absolute Galois group G := Gal(F / F)
14.9. LOCAL-GLOBAL PRINCIPLE
33
or just Gal(F), and to the extension F / Q. The fundamental theorem of Galois theory links fields in between F and its algebraic closure and closed subgroups of Gal (F). For example, the abelianization (the biggest abelian quotient) Gab of G corresponds to a field referred to as the maximal abelian extension F ab (called so since any further extension is not abelian, i.e., does not have an abelian Galois group). By the Kronecker–Weber theorem, the maximal abelian extension of Q is the extension generated by all roots of unity. For more general number fields, class field theory, specifically the Artin reciprocity law gives an answer by describing Gab in terms of the idele class group. Also notable is the Hilbert class field, the maximal abelian unramified field extension of F. It can be shown to be finite over F, its Galois group over F is isomorphic to the class group of F, in particular its degree equals the class number h of F (see above). In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is then also referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal(F), also known as Galois cohomology, which in the first place measures the failure of exactness of taking Gal(F)-invariants, but offers deeper insights (and questions) as well. For example, the Galois group G of a field extension L / F acts on L× , the nonzero elements of L. This Galois module plays a significant role in many arithmetic dualities, such as Poitou-Tate duality. The Brauer group of F, originally conceived to classify division algebras over F, can be recast as a cohomology group, namely H2 (Gal (F), F × ).
14.9 Local-global principle Generally speaking, the term “local to global” refers to the idea that a global problem is first done at a local level, which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be put together to get back to some global statement. For example, the notion of sheaves reifies that idea in topology and geometry.
14.9.1
Local and global fields
Number fields share a great deal of similarity with another class of fields much used in algebraic geometry known as function fields of algebraic curves over finite fields. An example is Fp(T). They are similar in many respects, for example in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient fields of which is the function field in question) of curves. Therefore, both types of field are called global fields. In accordance with the philosophy laid out above, they can be studied at a local level first, that is to say, by looking at the corresponding local fields. For number fields F, the local fields are the completions of F at all places, including the archimedean ones (see local analysis). For function fields, the local fields are completions of the local rings at all points of the curve for function fields. Many results valid for function fields also hold, at least if reformulated properly, for number fields. However, the study of number fields often poses difficulties and phenomena not encountered in function fields. For example, in function fields, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case.
14.9.2
Hasse principle
A prototypical question, posed at a global level, is whether some polynomial equation has a solution in F. If this is the case, this solution is also a solution in all completions. The local-global principle or Hasse principle asserts that for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solution can be done on all the completions of F, which is often easier, since analytic methods (classical analytic tools such as intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can be used. This implication does not hold, however, for more general types of equations. However, the idea of passing from local data to global ones proves fruitful in class field theory, for example, where local class field theory is used to obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completions Fᵥ can be explicitly determined, whereas the Galois groups of global fields, even of Q are far less understood.
34
CHAPTER 14. ALGEBRAIC NUMBER FIELD
14.9.3
Adeles and ideles
In order to assemble local data pertaining to all local fields attached to F, the adele ring is set up. A multiplicative variant is referred to as ideles.
14.10 See also • Dirichlet’s unit theorem, S-unit • Kummer extension • Minkowski’s theorem, Geometry of numbers • Chebotarev’s density theorem • Ray class group • Decomposition group • Genus field
14.11 Notes [1] Ireland, Kenneth; Rosen, Michael (1998), A Classical Introduction to Modern Number Theory, Berlin, New York: SpringerVerlag, ISBN 978-0-387-97329-6, Ch. 1.4 [2] Bloch, Spencer; Kato, Kazuya (1990), "L-functions and Tamagawa numbers of motives”, The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Boston, MA: Birkhäuser Boston, pp. 333–400, MR 1086888 [3] Narkiewicz 2004, §2.2.6 [4] Kleiner, Israel (1999), “Field theory: from equations to axiomatization. I”, The American Mathematical Monthly 106 (7): 677–684, doi:10.2307/2589500, MR 1720431, To Dedekind, then, fields were subsets of the complex numbers. [5] Mac Lane, Saunders (1981), “Mathematical models: a sketch for the philosophy of mathematics”, The American Mathematical Monthly 88 (7): 462–472, doi:10.2307/2321751, MR 628015, Empiricism sprang from the 19th-century view of mathematics as almost coterminal with theoretical physics. [6] Cohn [7] Conrad [8] Cohn
14.12 References • Cohn, Harvey (1988), A Classical Invitation to Algebraic Numbers and Class Fields, Universitext, New York: Springer-Verlag • Conrad, Keith http://www.math.uconn.edu/~{}kconrad/blurbs/gradnumthy/unittheorem.pdf • Janusz, Gerald J. (1996), Algebraic Number Fields (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0429-2 • Helmut Hasse, Number Theory, Springer Classics in Mathematics Series (2002) • Serge Lang, Algebraic Number Theory, second edition, Springer, 2000 • Richard A. Mollin, Algebraic Number Theory, CRC, 1999 • Ram Murty, Problems in Algebraic Number Theory, Second Edition, Springer, 2005
14.12. REFERENCES
35
• Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag, ISBN 978-3-540-21902-6, MR 2078267 • Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021 • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 1136.11001 • André Weil, Basic Number Theory, third edition, Springer, 1995
Chapter 15
Algebraic number theory Algebraic number theory is a major branch of number theory that studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field extensions. In this setting, the familiar features of the integers—such as unique factorization—need not hold. The virtue of the primary machinery employed—Galois theory, group cohomology, group representations, and L-functions—is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.
15.1 History of algebraic number theory 15.1.1
Diophantus
The beginnings of algebraic number theory can be traced to Diophantine equations,[1] named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:
A=x+y B = x2 + y 2 . Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).[2] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).[3] Diophantus’s major work was the Arithmetica, of which only a portion has survived.
15.1.2
Fermat
Fermat’s last theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century.
15.1.3
Gauss
One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae (Latin: Arithmetical Investigations) is a textbook of number theory written in Latin[4] by Carl Friedrich Gauss in 1798 when Gauss was 21 36
15.1. HISTORY OF ALGEBRAIC NUMBER THEORY
37
and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways. The Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular.
15.1.4
Dirichlet
In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms (later refined by his student Kronecker). The formula, which Jacobi called a result “touching the utmost of human acumen”, opened the way for similar results regarding more general number fields.[5] Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.[6] He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet’s approximation theorem. He published important contributions to Fermat’s last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law.[5] The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.
15.1.5
Dedekind
Richard Dedekind's study of Lejeune Dirichlet’s work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet’s lectures on number theory as Vorlesungen über Zahlentheorie (“Lectures on Number Theory”) about which it has been written that: “Although the book is assuredly based on Dirichlet’s lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet’s, the book itself was entirely written by Dedekind, for the most part after Dirichlet’s death.” (Edwards 1983) 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory. (The word “Ring”, introduced later by Hilbert, does not appear in Dedekind’s work.) Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer’s 1843 attempt to prove Fermat’s Last Theorem.
15.1.6
Hilbert
David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally “report on numbers”). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.[7] He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area. He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi.[8]
38
15.1.7
CHAPTER 15. ALGEBRAIC NUMBER THEORY
Artin
Emil Artin established the Artin reciprocity law in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory.[9] The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert’s product formula for the norm symbol. Artin’s result provided a partial solution to Hilbert’s ninth problem.
15.1.8
Modern theory
Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct, branches of mathematics, elliptic curves and modular forms. The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form. It was initially dismissed as unlikely or highly speculative, and was taken more seriously when number theorist André Weil found evidence supporting it, but no proof; as a result the “astounding”[10] conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the Langlands programme, a list of important conjectures needing proof or disproof. From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet’s theorem, provides a proof for Fermat’s Last Theorem. Both Fermat’s Last Theorem and the Modularity Theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians (meaning, impossible or virtually impossible to prove using current knowledge). Wiles first announced his proof in June 1993[11] in a version that was soon recognized as having a serious gap in a key point. The proof was corrected by Wiles, in part via collaboration with Richard Taylor, and the final, widely accepted, version was released in September 1994, and formally published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.
15.2 Basic notions 15.2.1
Unique factorization and the ideal class group
One of the first properties of Z that can fail in the ring of integers O of an algebraic number field K is that of the unique factorization of integers into prime numbers. The prime numbers in Z are generalized to irreducible elements in O, and though the unique factorization of elements of O into irreducible elements may hold in some cases (such as for the Gaussian integers Z[i]), it may also fail, as in the case of Z[√−5] where
6 = 2 · 3 = (1 +
√ √ −5) · (1 − −5).
The ideal class group of O is a measure of how much unique factorization of elements fails; in particular, the ideal class group is trivial if, and only if, O is a unique factorization domain.
15.2.2
Factoring prime ideals in extensions
Unique factorization can be partially recovered for O in that it has the property of unique factorization of ideals into prime ideals (i.e. it is a Dedekind domain). This makes the study of the prime ideals in O particularly important. This is another area where things change from Z to O: the prime numbers, which generate prime ideals of Z (in fact, every single prime ideal of Z is of the form (p):=pZ for some prime number p,) may no longer generate prime ideals in O. For example, in the ring of Gaussian integers, the ideal 2Z[i] is no longer a prime ideal; in fact
2
2Z[i] = ((1 + i)Z[i]) .
15.2. BASIC NOTIONS
39
On the other hand, the ideal 3Z[i] is a prime ideal. The complete answer for the Gaussian integers is obtained by using a theorem of Fermat’s, with the result being that for an odd prime number p
pZ[i] is a prime ideal if p ≡ 3 (mod 4) pZ[i] is not a prime ideal if p ≡ 1 (mod 4). Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when K is an abelian extension of Q (i.e. a Galois extension with abelian Galois group).
15.2.3
Primes and places
An important generalization of the notion of prime ideal in O is obtained by passing from the so-called ideal-theoretic approach to the so-called valuation-theoretic approach. The relation between the two approaches arises as follows. In addition to the usual absolute value function |·| : Q → R, there are absolute value functions |·| : Q → R defined for each prime number p in Z, called p-adic absolute values. Ostrowski’s theorem states that these are all possible absolute value functions on Q (up to equivalence). This suggests that the usual absolute value could be considered as another prime. More generally, a prime of an algebraic number field K (also called a place) is an equivalence class of absolute values on K. The primes in K are of two sorts: p -adic absolute values like |·| , one for each prime ideal p of O, and absolute values like |·| obtained by considering K as a subset of the complex numbers in various possible ways and using the absolute value |·| : C → R. A prime of the first kind is called a finite prime (or finite place) and one of the second kind is called an infinite prime (or infinite place). Thus, the set of primes of Q is generally denoted { 2, 3, 5, 7, ..., ∞ }, and the usual absolute value on Q is often denoted |·|∞ in this context. The set of infinite primes of K can be described explicitly in terms of the embeddings K → C (i.e. the non-zero ring homomorphisms from K to C). Specifically, the set of embeddings can be split up into two disjoint subsets, those whose image is contained in R, and the rest. To each embedding σ : K → R, there corresponds a unique prime of K coming from the absolute value obtained by composing σ with the usual absolute value on R; a prime arising in this fashion is called a real prime (or real place). To an embedding τ : K → C whose image is not contained in R, one can construct a distinct embedding τ, called the conjugate embedding, by composing τ with the complex conjugation map C → C. Given such a pair of embeddings τ and τ, there corresponds a unique prime of K again obtained by composing τ with the usual absolute value (composing τ instead gives the same absolute value function since |z| = |z| for any complex number z, where z denotes the complex conjugate of z). Such a prime is called a complex prime (or complex place). The description of the set of infinite primes is then as follows: each infinite prime corresponds either to a unique embedding σ : K → R, or a pair of conjugate embeddings τ, τ : K → C. The number of real (respectively, complex) primes is often denoted r1 (respectively, r2 ). Then, the total number of embeddings K → C is r1 +2r2 (which, in fact, equals the degree of the extension K/Q).
15.2.4
Units
The fundamental theorem of arithmetic describes the multiplicative structure of Z. It states that every non-zero integer can be written (essentially) uniquely as a product of prime powers and ±1. The unique factorization of ideals in the ring O recovers part of this description, but fails to address the factor ±1. The integers 1 and −1 are the invertible elements (i.e. units) of Z. More generally, the invertible elements in O form a group under multiplication called the unit group of O, denoted O× . This group can be much larger than the cyclic group of order 2 formed by the units of Z. Dirichlet’s unit theorem describes the abstract structure of the unit group as an abelian group. A more precise statement giving the structure of O× ⊗Z Q as a Galois module for the Galois group of K/Q is also possible.[12] The size of the unit group, and its lattice structure give important numerical information about O, as can be seen in the class number formula.
15.2.5
Local fields
Main article: Local field
40
CHAPTER 15. ALGEBRAIC NUMBER THEORY
Completing a number field K at a place w gives a complete field. If the valuation is archimedean, one gets R or C, if it is non-archimedean and lies over a prime p of the rationals, one gets a finite extension K / Q : a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example the Kronecker–Weber theorem can be deduced easily from the analogous local statement. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry.
15.3 Major results 15.3.1
Finiteness of the class group
One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field K is finite. The order of the class group is called the class number, and is often denoted by the letter h.
15.3.2
Dirichlet’s unit theorem
Main article: Dirichlet’s unit theorem Dirichlet’s unit theorem provides a description of the structure of the multiplicative group of units O× of the ring of integers O. Specifically, it states that O× is isomorphic to G × Zr , where G is the finite cyclic group consisting of all the roots of unity in O, and r = r1 + r2 − 1 (where r1 (respectively, r2 ) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of K). In other words, O× is a finitely generated abelian group of rank r1 + r2 − 1 whose torsion consists of the roots of unity in O.
15.3.3
Reciprocity laws
Main article: Reciprocity law In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states ( )( ) p−1 q−1 p q = (−1) 2 2 . q p A reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see. See also Quadratic reciprocity Cubic reciprocity Quartic reciprocity Artin reciprocity law
15.4. RELATED AREAS
15.3.4
41
Class number formula
Main article: Class number formula The class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
15.4 Related areas Algebraic number theory interacts with many other mathematical disciplines. It uses tools from homological algebra. Via the analogy of function fields vs. number fields, it relies on techniques and ideas from algebraic geometry. Moreover, the study of higher-dimensional schemes over Z instead of number rings is referred to as arithmetic geometry. Algebraic number theory is also used in the study of arithmetic hyperbolic 3-manifolds.
15.5 See also • Langlands program • Adele ring • Tamagawa number • Iwasawa theory • Arithmetic algebraic geometry • Class field theory • Kummer theory • Ideal class group
15.6 Notes [1] Stark, pp. 145–146. [2] Aczel, pp. 14–15. [3] Stark, pp. 44–47. [4] Disquisitiones Arithmeticae at Yalepress.yale.edu [5] Elstrodt, Jürgen (2007). “The Life and Work of Gustav Lejeune Dirichlet (1805–1859)" (PDF). Clay Mathematics Proceedings. Retrieved 2007-12-25. [6] Kanemitsu, Shigeru; Chaohua Jia (2002). Number theoretic methods: future trends. Springer. pp. 271–274. ISBN 978-14020-1080-4. [7] Reid, Constance, 1996. Hilbert, Springer, ISBN 0-387-94674-8. [8] This work established Takagi as Japan’s first mathematician of international stature. [9] Helmut Hasse, History of Class Field Theory, in Algebraic Number Theory, edited by Cassels and Frölich, Academic Press, 1967, pp. 266–279 [10] Fermat’s Last Theorem, Simon Singh, 1997, ISBN 1-85702-521-0> [11] Kolata, Gina (24 June 1993). “At Last, Shout of 'Eureka!' In Age-Old Math Mystery”. The New York Times. Retrieved 21 January 2013. [12] See proposition VIII.8.6.11 of Neukirch, Schmidt & Wingberg 2000
42
CHAPTER 15. ALGEBRAIC NUMBER THEORY • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, Zbl 0948.11001, MR 1737196
15.7 Further reading 15.7.1
Introductory texts
• Kenneth Ireland and Michael Rosen, “A Classical Introduction to Modern Number Theory, Second Edition”, Springer-Verlag, 1990 • Ian Stewart and David O. Tall, “Algebraic Number Theory and Fermat’s Last Theorem,” A. K. Peters, 2002
15.7.2
Intermediate texts
• Daniel A. Marcus, “Number Fields”
15.7.3
Graduate level accounts
• Cassels, J. W. S.; Fröhlich, Albrecht, eds. (1967), Algebraic number theory, London: Academic Press, MR 0215665 • Fröhlich, Albrecht; Taylor, Martin J. (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, ISBN 0-521-43834-9, MR 1215934 • Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics 110 (2 ed.), New York: SpringerVerlag, ISBN 978-0-387-94225-4, MR 1282723 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
15.8 External links • Hazewinkel, Michiel, ed. (2001), “Algebraic number theory”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
15.8. EXTERNAL LINKS
Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory.
43
Chapter 16
Algebraically closed field In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x], the ring of polynomials in the variable x with coefficients in F.
16.1 Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field F is algebraically closed, because if a1 , a2 , …, an are the elements of F, then the polynomial (x − a1 )(x − a2 ) ··· (x − an) + 1 has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of (complex) algebraic numbers.
16.2 Equivalent properties Given a field F, the assertion "F is algebraically closed” is equivalent to other assertions:
16.2.1
The only irreducible polynomials are those of degree one
The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F[x] are those of degree one. The assertion “the polynomials of degree one are irreducible” is trivially true for any field. If F is algebraically closed and p(x) is an irreducible polynomial of F[x], then it has some root a and therefore p(x) is a multiple of x − a. Since p(x) is irreducible, this means that p(x) = k(x − a), for some k ∈ F \ {0}. On the other hand, if F is not algebraically closed, then there is some non-constant polynomial p(x) in F[x] without roots in F. Let q(x) be some irreducible factor of p(x). Since p(x) has no roots in F, q(x) also has no roots in F. Therefore, q(x) has degree greater than one, since every first degree polynomial has one root in F.
16.2.2
Every polynomial is a product of first degree polynomials
The field F is algebraically closed if and only if every polynomial p(x) of degree n ≥ 1, with coefficients in F, splits into linear factors. In other words, there are elements k, x1 , x2 , …, xn of the field F such that p(x) = k(x − x1 )(x − x2 ) ··· (x − xn). If F has this property, then clearly every non-constant polynomial in F[x] has some root in F; in other words, F is algebraically closed. On the other hand, that the property stated here holds for F if F is algebraically closed follows from the previous property together with the fact that, for any field K, any polynomial in K[x] can be written as a product of irreducible polynomials. 44
16.2. EQUIVALENT PROPERTIES
16.2.3
45
Polynomials of prime degree have roots
J. Shipman showed in 2007 that if every polynomial over F of prime degree has a root in F, then every non-constant polynomial has a root in F, thus F is algebraically closed.
16.2.4
The field has no proper algebraic extension
The field F is algebraically closed if and only if it has no proper algebraic extension. If F has no proper algebraic extension, let p(x) be some irreducible polynomial in F[x]. Then the quotient of F[x] modulo the ideal generated by p(x) is an algebraic extension of F whose degree is equal to the degree of p(x). Since it is not a proper extension, its degree is 1 and therefore the degree of p(x) is 1. On the other hand, if F has some proper algebraic extension K, then the minimal polynomial of an element in K \ F is irreducible and its degree is greater than 1.
16.2.5
The field has no proper finite extension
The field F is algebraically closed if and only if it has no finite algebraic extension because if, within the previous proof, the word “algebraic” is replaced by the word “finite”, then the proof is still valid.
16.2.6
Every endomorphism of Fn has some eigenvector
The field F is algebraically closed if and only if, for each natural number n, every linear map from Fn into itself has some eigenvector. An endomorphism of Fn has an eigenvector if and only if its characteristic polynomial has some root. Therefore, when F is algebraically closed, every endomorphism of Fn has some eigenvector. On the other hand, if every endomorphism of Fn has an eigenvector, let p(x) be an element of F[x]. Dividing by its leading coefficient, we get another polynomial q(x) which has roots if and only if p(x) has roots. But if q(x) = xn + an ₋ ₁xn − 1 + ··· + a0 , then q(x) is the characteristic polynomial of the companion matrix
0 0 1 0 0 1 .. .. . . 0 0
16.2.7
··· ··· ··· .. .
0 0 0 .. .
−a0 −a1 −a2 .. .
···
1
−an−1
.
Decomposition of rational expressions
The field F is algebraically closed if and only if every rational function in one variable x, with coefficients in F, can be written as the sum of a polynomial function with rational functions of the form a/(x − b)n , where n is a natural number, and a and b are elements of F. If F is algebraically closed then, since the irreducible polynomials in F[x] are all of degree 1, the property stated above holds by the theorem on partial fraction decomposition. On the other hand, suppose that the property stated above holds for the field F. Let p(x) be an irreducible element in F[x]. Then the rational function 1/p can be written as the sum of a polynomial function q with rational functions of the form a/(x − b)n . Therefore, the rational expression 1 − p(x)q(x) 1 − q(x) = p(x) p(x) can be written as a quotient of two polynomials in which the denominator is a product of first degree polynomials. Since p(x) is irreducible, it must divide this product and, therefore, it must also be a first degree polynomial.
46
CHAPTER 16. ALGEBRAICALLY CLOSED FIELD
16.2.8
Relatively prime polynomials and roots
For any field F, if two polynomials p(x),q(x) ∈ F[x] are relatively prime then they do not have a common root, for if a ∈ F was a common root, then p(x) and q(x) would both be multiples of x − a and therefore they would not be relatively prime. The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields. If the field F is algebraically closed, let p(x) and q(x) be two polynomials which are not relatively prime and let r(x) be their greatest common divisor. Then, since r(x) is not constant, it will have some root a, which will be then a common root of p(x) and q(x). If F is not algebraically closed, let p(x) be a polynomial whose degree is at least 1 without roots. Then p(x) and p(x) are not relatively prime, but they have no common roots (since none of them has roots).
16.3 Other properties If F is an algebraically closed field and n is a natural number, then F contains all nth roots of unity, because these are (by definition) the n (not necessarily distinct) zeroes of the polynomial xn − 1. A field extension that is contained in an extension generated by the roots of unity is a cyclotomic extension, and the extension of a field generated by all roots of unity is sometimes called its cyclotomic closure. Thus algebraically closed fields are cyclotomically closed. The converse is not true. Even assuming that every polynomial of the form xn − a splits into linear factors is not enough to assure that the field is algebraically closed. If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is it valid for all other algebraically closed fields with characteristic 0, but there is some natural number N such that the proposition is valid for every algebraically closed field with characteristic p when p > N.[1] Every field F has some extension which is algebraically closed. Among all such extensions there is one and (up to isomorphism, but not unique isomorphism) only one which is an algebraic extension of F;[2] it is called the algebraic closure of F. The theory of algebraically closed fields has quantifier elimination.
16.4 Notes [1] See subsections Rings and fields and Properties of mathematical theories in §2 of J. Barwise’s “An introduction to first-order logic”. [2] See Lang’s Algebra, §VII.2 or van der Waerden’s Algebra I, §10.1.
16.5 References • Barwise, Jon (1978), “An introduction to first-order logic”, in Barwise, Jon, Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, North Holland, ISBN 0-7204-2285-X • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: SpringerVerlag, ISBN 978-0-387-95385-4, MR 1878556 • Shipman, Joseph (2007), “Improving the Fundamental Theorem of Algebra”, Mathematical Intelligencer 29 (4): 9–14, doi:10.1007/BF02986170, ISSN 0343-6993 • van der Waerden, Bartel Leendert (2003), Algebra I (7th ed.), Springer-Verlag, ISBN 0-387-40624-7
Chapter 17
All one polynomial An all one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are known, which allow this polynomial to be used to define efficient algorithms and circuits for multiplication in finite fields of characteristic two.[1] The AOP is a 1-equally spaced polynomial.[2]
17.1 Definition An AOP of degree m has all terms from xm to x0 with coefficients of 1, and can be written as
AOPm (x) =
m ∑
xi
i=0
or
AOPm (x) = xm + xm−1 + · · · + x + 1 or
AOPm (x) =
xm+1 − 1 x−1
thus the roots of the all one polynomial of degree m are all (m+1)th roots of unity other than unity itself.
17.2 Properties Over GF(2) the AOP has many interesting properties, including: • The Hamming weight of the AOP is m + 1, the maximum possible for its degree[3] • The AOP is irreducible if and only if m + 1 is prime and 2 is a primitive root modulo m + 1[1] • The only AOP that is a primitive polynomial is x2 + x + 1. Despite the fact that the Hamming weight is large, because of the ease of representation and other improvements there are efficient implementations in areas such as coding theory and cryptography.[1] Over Q , the AOP is irreducible whenever m + 1 is prime p, and therefore in these cases, the pth cyclotomic polynomial.[4] 47
48
CHAPTER 17. ALL ONE POLYNOMIAL
17.3 References [1] Cohen, Henri; Frey, Gerhard; Avanzi, Roberto; Doche, Christophe; Lange, Tanja; Nguyen, Kim; Vercauteren, Frederik (2005), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Discrete Mathematics and Its Applications, CRC Press, p. 215, ISBN 9781420034981. [2] Itoh, Toshiya; Tsujii, Shigeo (1989), “Structure of parallel multipliers for a class of fields GF(2m )", Information and Computation 83 (1): 21–40, doi:10.1016/0890-5401(89)90045-X. [3] Reyhani-Masoleh, Arash; Hasan, M. Anwar (2003), “On low complexity bit parallel polynomial basis multipliers”, Cryptographic Hardware and Embedded Systems - CHES 2003, Lecture Notes in Computer Science 2779, Springer, pp. 189–202, doi:10.1007/978-3-540-45238-6_16. [4] Sugimura, Tatsuo; Suetugu, Yasunori (1991), “Considerations on irreducible cyclotomic polynomials”, Electronics and Communications in Japan 74 (4): 106–113, doi:10.1002/ecjc.4430740412, MR 1136200.
17.4 External links • all one polynomial at PlanetMath.org.
Chapter 18
Archimedean property
Illustration of the Archimedean property.
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder.[1] The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields. An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group. 49
50
CHAPTER 18. ARCHIMEDEAN PROPERTY
This can be made precise in various contexts with slightly different ways of formulation. For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.
18.1 History and origin of the name of the Archimedean property The concept was named by Otto Stolz (in the 1880s) after the ancient Greek geometer and physicist Archimedes of Syracuse. The Archimedean property appears in Book V of Euclid’s Elements as Definition 4: Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. Because Archimedes credited it to Eudoxus of Cnidus it is also known as the “Theorem of Eudoxus”[2] or the Eudoxus axiom. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.
18.2 Definition for linearly ordered groups Let x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y (or equivalently, y is infinite with respect to x) if, for every natural number n, the multiple nx is less than y, that is, the following inequality holds:
x + · · · + x < y. | {z } nterms
The group G is Archimedean if there is no pair x,y such that x is infinitesimal with respect to y. Additionally, if K is an algebraic structure with a unit (1) — for example, a ring — a similar definition applies to K. If x is infinitesimal with respect to 1, then x is an infinitesimal element. Likewise, if y is infinite with respect to 1, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements and no infinitesimal elements.
18.2.1
Ordered fields
An ordered field has some additional properties. • One may assume that the rational numbers are contained in the field. • If x is infinitesimal, then 1/x is infinite, and vice versa. Therefore to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. • If x is infinitesimal and r is a rational number, then r x is also infinitesimal. As a result, given a general element c, the three numbers c/2, c, and 2c are either all infinitesimal or all non-infinitesimal. In this setting, an ordered field K is Archimedean precisely when the following statement, called the axiom of Archimedes, holds: Let x be any element of K. Then there exists a natural number n such that n > x. Alternatively one can use the following characterization: For any positive ε in K, there exists a natural number n, such that 1/n < ε.
18.3. DEFINITION FOR NORMED FIELDS
51
18.3 Definition for normed fields The qualifier “Archimedean” is also formulated in the theory of rank one valued fields and normed spaces over rank one valued fields as follows. Let F be a field endowed with an absolute value function, i.e., a function which associates the real number 0 with the field element 0 and associates a positive real number |x| with each non-zero x ∈ F and satisfies |xy| = |x||y| and |x + y| ≤ |x| + |y| . Then, F is said to be Archimedean if for any non-zero x ∈ F there exists a natural number n such that
| x + · · · + x | > 1. | {z } nterms
Similarly, a normed space is Archimedean if a sum of n terms, each equal to a non-zero vector x , has norm greater than one for sufficiently large n . A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the ultrametric triangle inequality,
|x + y| ≤ max(|x|, |y|) respectively. A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean. The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.[3]
18.4 Examples and non-examples 18.4.1
Archimedean property of the real numbers
The field of the rational numbers can be assigned one√of a number of absolute value functions, including the trivial function |x| = 1, when x ̸= 0 , the more usual |x| = x2 , and the p-adic absolute value functions. By Ostrowski’s theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some p-adic absolute value. The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete. The completion with respect to the usual absolute value (from the order) is the field of real numbers. By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field. [4] On the other hand, the completions with respect to the other non-trivial absolute values give the fields of p-adic numbers, where p is a prime integer number (see below); since the p-adic absolute values satisfy the ultrametric property, then the p-adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields). In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property as follows. Denote by Z the set consisting of all positive infinitesimals. This set is bounded above by 1. Now assume for a contradiction that Z is nonempty. Then it has a least upper bound c, which is also positive, so c/2 < c < 2c. Since c is an upper bound of Z and 2c is strictly larger than c, 2c is not a positive infinitesimal. That is, there is some natural number n for which 1/n < 2c. On the other hand, c/2 is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimal x between c/2 and c, and if 1/k < c/2 <= x then x is not infinitesimal. But 1/(4n) < c/2, so c/2 is not infinitesimal, and this is a contradiction. This means that Z is empty after all: there are no positive, infinitesimal real numbers. One should note that the Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context.
18.4.2
Non-Archimedean ordered field
Main article: Non-Archimedean ordered field
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CHAPTER 18. ARCHIMEDEAN PROPERTY
For an example of an ordered field that is not Archimedean, take the field of rational functions with real coefficients. (A rational function is any function that can be expressed as one polynomial divided by another polynomial; we will assume in what follows that this has been done in such a way that the leading coefficient of the denominator is positive.) To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations. Now f > g if and only if f − g > 0, so we only have to say which rational functions are considered positive. Call the function positive if the leading coefficient of the numerator is positive. (One must check that this ordering is well defined and compatible with addition and multiplication.) By this definition, the rational function 1/x is positive but less than the rational function 1. In fact, if n is any natural number, then n(1/x) = n/x is positive but still less than 1, no matter how big n is. Therefore, 1/x is an infinitesimal in this field. This example generalizes to other coefficients. Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field. Taking the coefficients to be the rational functions in a different variable, say y, produces an example with a different order type.
18.4.3
Non-Archimedean valued fields
The field of the rational numbers endowed with the p-adic metric and the p-adic number fields which are the completions, do not have the Archimedean property as fields with absolute values. All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value.[5] There is a non-trivial non-Archimedean valuation on every infinite field.
18.4.4
Equivalent definitions of Archimedean ordered field
Every linearly ordered field K contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of K, which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered monoid. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in K. The following are equivalent characterizations of Archimedean fields in terms of these substructures.[6] 1. The natural numbers are cofinal in K. That is, every element of K is less than some natural number. (This is not the case when there exist infinite elements.) Thus an Archimedean field is one whose natural numbers grow without bound. 2. Zero is the infimum in K of the set {1/2, 1/3, 1/4, … }. (If K contained a positive infinitesimal it would be a lower bound for the set whence zero would not be the greatest lower bound.) 3. The set of elements of K between the positive and negative rationals is closed. This is because the set consists of all the infinitesimals, which is just the closed set {0} when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. In the latter case, (i) every infinitesimal is less than every positive rational, (ii) there is neither a greatest infinitesimal nor a least positive rational, and (iii) there is nothing else in between, a situation that points up both the incompleteness and disconnectedness of any non-Archimedean field. 4. For any x in K the set of integers greater than x has a least element. (If x were a negative infinite quantity every integer would be greater than it.) 5. Every nonempty open interval of K contains a rational. (If x is a positive infinitesimal, the open interval (x, 2x) contains infinitely many infinitesimals but not a single rational.) 6. The rationals are dense in K with respect to both sup and inf. (That is, every element of K is the sup of some set of rationals, and the inf of some other set of rationals.) Thus an Archimedean field is any dense ordered extension of the rationals, in the sense of any ordered field that densely embeds its rational elements.
18.5 Notes [1] G. Fisher (1994) in P. Ehrlich(ed.), Real Numbers, Generalizations of the Reals, and Theories of continua, 107-145, Kluwer Academic [2] Knopp, Konrad (1951). Theory and Application of Infinite Series (English 2nd ed.). London and Glasgow: Blackie & Son, Ltd. p. 7. ISBN 0-486-66165-2. [3] Monna, A. F., Over een lineare P-adisches ruimte, Indag. Math., 46 (1943), 74–84.
18.6. REFERENCES
53
[4] Neal Koblitz, “p-adic Numbers, p-adic Analysis, and Zeta-Functions”, Springer-Verlag,1977. [5] Shell, Niel, Topological Fields and Near Valuations, Dekker, New York, 1990. ISBN 0-8247-8412-X [6] Schechter 1997, §10.3
18.6 References • Schechter, Eric (1997). Handbook of Analysis and its Foundations. Academic Press. ISBN 0-12-622760-8.
Chapter 19
Arithmetic and geometric Frobenius In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping φ that takes r in R to rp is a ring endomorphism of R. The image of φ is then Rp , the subring of R consisting of p-th powers. In some important cases, for example finite fields, φ is surjective. Otherwise φ is an endomorphism but not a ring automorphism. The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to φ. This gives a mapping φ*: Spec(Rp ) → Spec(R) of affine schemes. Even in cases where Rp = R this is not the identity, unless R is the prime field. Mappings created by fibre product with φ*, i.e. base changes, tend in scheme theory to be called geometric Frobenius. The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structure, is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear.
19.1 References • Freitag, Eberhard; Kiehl, Reinhardt (1988), Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 13, Berlin, New York: Springer-Verlag, ISBN 978-3-540-12175-6, MR 926276, p. 5
54
Chapter 20
Arithmetic dynamics Arithmetic dynamics[1] is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. Global arithmetic dynamics refers to the study of analogues of classical Diophantine geometry in the setting of discrete dynamical systems, while local arithmetic dynamics, also called p-adic or nonarchimedean dynamics, is an analogue of classical dynamics in which one replaces the complex numbers C by a p-adic field such as Qp or Cp and studies chaotic behavior and the Fatou and Julia sets. The following table describes a rough correspondence between Diophantine equations, especially abelian varieties, and dynamical systems:
20.1 Definitions and notation from discrete dynamics Let S be a set and let F : S → S be a map from S to itself. The iterate of F with itself n times is denoted F (n) = F ◦ F ◦ · · · ◦ F. A point P ∈ S is periodic if F (n) (P) = P for some n > 1. The point is preperiodic if F (k) (P) is periodic for some k ≥ 1. The (forward) orbit of P is the set { } OF (P ) = P, F (P ), F (2) (P ), F (3) (P ), · · · . Thus P is preperiodic if and only if its orbit OF(P) is finite.
20.2 Number theoretic properties of preperiodic points Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of Northcott[2] says that F has only finitely many Q-rational preperiodic points, i.e., F has only finitely many preperiodic points in P1 (Q). The Uniform Boundedness Conjecture[3] of Morton and Silverman says that the number of preperiodic points of F in P1 (Q) is bounded by a constant that depends only on the degree of F. More generally, let F : PN → PN be a morphism of degree at least two defined over a number field K. Northcott’s theorem says that F has only finitely many preperiodic points in PN (K), and the general Uniform Boundedness Conjecture says that the number of preperiodic points in PN (K) may be bounded solely in terms of N, the degree of F, and the degree of K over Q. 55
56
CHAPTER 20. ARITHMETIC DYNAMICS
The Uniform Boundedness Conjecture is not known even for quadratic polynomials Fc(x) = x2 + c over the rational numbers Q. It is known in this case that Fc(x) cannot have periodic points of period four,[4] five,[5] or six,[6] although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer. Poonen has conjectured that Fc(x) cannot have rational periodic points of any period strictly larger than three.[7]
20.3 Integer points in orbits The orbit of a rational map may contain infinitely many integers. For example, if F(x) is a polynomial with integer coefficients and if a is an integer, then it is clear that the entire orbit OF(a) consists of integers. Similarly, if F(x) is a rational map and some iterate F (n) (x) is a polynomial with integer coefficients, then every n-th entry in the orbit is an integer. An example of this phenomenon is the map F(x) = x−d , whose second iterate is a polynomial. It turns out that this is the only way that an orbit can contain infinitely many integers. Theorem.[8] Let F(x) ∈ Q(x) be a rational function of degree at least two, and assume that no iterate[9] of F is a polynomial. Let a ∈ Q. Then the orbit OF(a) contains only finitely many integers.
20.4 Dynamically defined points lying on subvarieties There are general conjectures due to Shouwu Zhang[10] and others concerning subvarieties that contain infinitely many periodic points or that intersect an orbit in infinitely many points. These are dynamical analogues of, respectively, the Manin–Mumford conjecture, proven by Raynaud, and the Mordell–Lang conjecture, proven by Faltings. The following conjectures illustrate the general theory in the case that the subvariety is a curve. Conjecture. Let F : PN → PN be a morphism and let C ⊂ PN be an irreducible algebraic curve. Suppose that either of the following is true: (a) C contains infinitely many points that are periodic points of F. (b) There is a point P ∈ PN such that C contains infinitely many points in the orbit OF(P). Then C is periodic for F in the sense that there is some iterate F (k) of F that maps C to itself.
20.5 p-adic dynamics The field of p-adic (or nonarchimedean) dynamics is the study of classical dynamical questions over a field K that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the field of p-adic rationals Qp and the completion of its algebraic closure Cp. The metric on K and the standard definition of equicontinuity leads to the usual definition of the Fatou and Julia sets of a rational map F(x) ∈ K(x). There are many similarities between the complex and the nonarchimedean theories, but also many differences. A striking difference is that in the nonarchimedean setting, the Fatou set is always nonempty, but the Julia set may be empty. This is the reverse of what is true over the complex numbers. Nonarchimedean dynamics has been extended to Berkovich space,[11] which is a compact connected space that contains the totally disconnected non-locally compact field Cp.
20.6 Generalizations There are natural generalizations of arithmetic dynamics in which Q and Qp are replaced by number fields and their p-adic completions. Another natural generalization is to replace self-maps of P1 or PN with self-maps (morphisms) V → V of other affine or projective varieties.
20.7 Other areas in which number theory and dynamics interact There are many other problems of a number theoretic nature that appear in the setting of dynamical systems, including:
20.8. SEE ALSO
57
• dynamics over finite fields. • dynamics over function fields such as C(x). • iteration of formal and p-adic power series. • dynamics on Lie groups. • arithmetic properties of dynamically defined moduli spaces. • equidistribution[12] and invariant measures, especially on p-adic spaces. • dynamics on Drinfeld modules. • number-theoretic iteration problems that are not described by rational maps on varieties, for example, the Collatz problem. • symbolic codings of dynamical systems based on explicit arithmetic expansions of real numbers.[13] The Arithmetic Dynamics Reference List gives an extensive list of articles and books covering a wide range of arithmetical dynamical topics.
20.8 See also • Arithmetic geometry • Arithmetic topology • Combinatorics and dynamical systems
20.9 Notes and references [1] J.H. Silverman (2007). The Arithmetic of Dynamical Systems. Springer. ISBN 978-0-387-69903-5. [2] D. G. Northcott. Periodic points on an algebraic variety. Ann. of Math. (2), 51:167-−177, 1950. [3] P. Morton and J. H. Silverman. Rational periodic points of rational functions. Internat. Math. Res. Notices, (2):97-−110, 1994. [4] P. Morton. Arithmetic properties of periodic points of quadratic maps. Acta Arith., 62(4):343-−372, 1992. [5] E. V. Flynn, B. Poonen, and E. F. Schaefer. Cycles of quadratic polynomials and rational points on a genus-2 curve. Duke Math. J., 90(3):435-−463, 1997. [6] M. Stoll, Rational 6-cycles under iteration of quadratic polynomials, 2008. [7] B. Poonen. The classification of rational preperiodic points of quadratic polynomials over Q: a refined conjecture. Math. Z., 228(1):11-−29, 1998. [8] J. H. Silverman. Integer points, Diophantine approximation, and iteration of rational maps. Duke Math. J., 71(3):793-829, 1993. [9] An elementary theorem says that if F(x) ∈ C(x) and if some iterate of F is a polynomial, then already the second iterate is a polynomial. [10] S.-W. Zhang, Distributions in algebraic dynamics, Differential Geometry: A Tribute to Professor S.-S. Chern, Surv. Differ. Geom., Vol. X, Int. Press, Boston, MA, 2006, pages 381–430. [11] R. Rumely and M. Baker, Analysis and dynamics on the Berkovich projective line, ArXiv preprint, 150 pages. [12] Equidistribution in number theory, an introduction, Andrew Granville, Zeév Rudnick Springer, 2007, ISBN 978-1-40205403-7 [13] Sidorov, Nikita (2003). “Arithmetic dynamics”. In Bezuglyi, Sergey; Kolyada, Sergiy. Topics in dynamics and ergodic theory. Survey papers and mini-courses presented at the international conference and US-Ukrainian workshop on dynamical systems and ergodic theory, Katsiveli, Ukraine, August 21–30, 2000. Lond. Math. Soc. Lect. Note Ser. 310. Cambridge: Cambridge University Press. pp. 145–189. ISBN 0-521-53365-1. Zbl 1051.37007.
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20.10 Further reading • Lecture Notes on Arithmetic Dynamics Arizona Winter School, March 13–17, 2010, Joseph H. Silverman • Chapter 15 of A first course in dynamics: with a panorama of recent developments, Boris Hasselblatt, A. B. Katok, Cambridge University Press, 2003, ISBN 978-0-521-58750-1
20.11 External links • The Arithmetic of Dynamical Systems home page • Arithmetic dynamics bibliography • Analysis and dynamics on the Berkovich projective line • Book review of Joseph H. Silverman's “The Arithmetic of Dynamical Systems”, reviewed by Robert L. Benedetto
Chapter 21
Artin L-function In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complexanalytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and Langlands’ philosophy. So far, only a small part of such a theory has been put on a firm basis.
21.1 Definition Given ρ , a representation of G on a finite-dimensional complex vector space V , where G is the Galois group of the finite extension L/K of number fields, the Artin L -function: L(ρ, s) is defined by an Euler product. For each prime ideal p in K 's ring of integers, there is an Euler factor, which is easiest to define in the case where p is unramified in L (true for almost all p ). In that case, the Frobenius element Frob(p) is defined as a conjugacy class in G . Therefore the characteristic polynomial of ρ(Frob(p)) is well-defined. The Euler factor for p is a slight modification of the characteristic polynomial, equally well-defined, charpoly(ρ(Frob(p)))−1 = det [I − tρ(Frob(p))] −s
as rational function in t, evaluated at t = N (p) notation. (Here N is the field norm of an ideal.)
−1
,
, with s a complex variable in the usual Riemann zeta function
When p is ramified, and I is the inertia group which is a subgroup of G, a similar construction is applied, but to the subspace of V fixed (pointwise) by I.[note 1] The Artin L-function L(ρ, s) is then the infinite product over all prime ideals p of these factors. As Artin reciprocity shows, when G is an abelian group these L-functions have a second description (as Dirichlet L-functions when K is the rational number field, and as Hecke L-functions in general). Novelty comes in with non-abelian G and their representations. One application is to give factorisations of Dedekind zeta-functions, for example in the case of a number field that is Galois over the rational numbers. In accordance with the decomposition of the regular representation into irreducible representations, such a zeta-function splits into a product of Artin L-functions, for each irreducible representation of G. For example, the simplest case is when G is the symmetric group on three letters. Since G has an irreducible representation of degree 2, an Artin L-function for such a representation occurs, squared, in the factorisation of the Dedekind zeta-function for such a number field, in a product with the Riemann zeta-function (for the trivial representation) and an L-function of Dirichlet’s type for the signature representation.
21.2 Functional equation Artin L-functions satisfy a functional equation. The function L(ρ,s) is related in its values to L(ρ*, 1 − s), where ρ* denotes the complex conjugate representation. More precisely L is replaced by Λ(ρ, s), which is L multiplied by 59
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CHAPTER 21. ARTIN L-FUNCTION
certain gamma factors, and then there is an equation of meromorphic functions Λ(ρ, s) = W(ρ)Λ(ρ*, 1 − s) with a certain complex number W(ρ) of absolute value 1. It is the Artin root number. It has been studied deeply with respect to two types of properties. Firstly Langlands and Deligne established a factorisation into Langlands– Deligne local constants; this is significant in relation to conjectural relationships to automorphic representations. Also the case of ρ and ρ* being equivalent representations is exactly the one in which the functional equation has the same L-function on each side. It is, algebraically speaking, the case when ρ is a real representation or quaternionic representation. The Artin root number is, then, either +1 or −1. The question of which sign occurs is linked to Galois module theory (Perlis 2001).
21.3 The Artin conjecture The Artin conjecture on Artin L-functions states that the Artin L-function L(ρ,s) of a non-trivial irreducible representation ρ is analytic in the whole complex plane.[1] This is known for one-dimensional representations, the L-functions being then associated to Hecke characters — and in particular for Dirichlet L-functions.[1] More generally Artin showed that the Artin conjecture is true for all representations induced from 1-dimensional representations. If the Galois group is supersolvable then all representations are of this form so the Artin conjecture holds. André Weil proved the Artin conjecture in the case of function fields. Two dimensional representations are classified by the nature of the image subgroup: it may be cyclic, dihedral, tetrahedral, octahedral, or icosahedral. The Artin conjecture for the cyclic or dihedral case follows easily from Hecke’s work. Langlands used the base change lifting to prove the tetrahedral case, and Tunnell extended his work to cover the octahedral case; Wiles used these cases in his proof of the Taniyama–Shimura conjecture. Richard Taylor and others have made some progress on the (non-solvable) icosahedral case; this is an active area of research. Brauer’s theorem on induced characters implies that all Artin L-functions are products of positive and negative integral powers of Hecke L-functions, and are therefore meromorphic in the whole complex plane. Langlands (1970) pointed out that the Artin conjecture follows from strong enough results from the Langlands philosophy, relating to the L-functions associated to automorphic representations for GL(n) for all n ≥ 1 . More precisely, the Langlands conjectures associate an automorphic representation of the adelic group GL (AQ) to every n-dimensional irreducible representation of the Galois group, which is a cuspidal representation if the Galois representation is irreducible, such that the Artin L-function of the Galois representation is the same as the automorphic L-function of the automorphic representation. The Artin conjecture then follows immediately from the known fact that the L-functions of cuspidal automorphic representations are holomorphic. This was one of the major motivations for Langlands’ work.
21.4 See also • Equivariant L-function
21.5 Notes [1] It is arguably more correct to think instead about the coinvariants, the largest quotient space fixed by I, rather than the invariants, but the result here will be the same. Cf. Hasse–Weil L-function for a similar situation.
21.6 References [1] Martinet (1977) p.18
21.7. EXTERNAL LINKS
61
• Artin, E. (1923). "Über eine neue Art von L Reihen". Hamb. Math. Abh. 3. Reprinted in his collected works, ISBN 0-387-90686-X. English translation in Artin L-Functions: A Historical Approach by N. Snyder. • Artin, Emil (1930), “Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren.”, Abhandlungen Hamburg (in German) 8: 292–306, doi:10.1007/BF02941010, JFM 56.0173.02 • Tunnell, Jerrold (1981). “Artin’s conjecture for representations of octahedral type”. Bull. Amer. Math. Soc. N. S. 5 (2): 173–175. doi:10.1090/S0273-0979-1981-14936-3. • Gelbart, Stephen (1977). “Automorphic forms and Artin’s conjecture”. Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn., Bonn, 1976). Lecture Notes in Math. 627. Berlin: Springer. pp. 241–276. • Langlands, Robert (1967), Letter to Prof. Weil • Langlands, R. P. (1970), “Problems in the theory of automorphic forms”, Lectures in modern analysis and applications, III, Lecture Notes in Math 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614 • Martinet, J. (1977), “Character theory and Artin L-functions”, in Fröhlich, A., Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975, Academic Press, pp. 1–87, ISBN 0-12-268960-7, Zbl 0359.12015
21.7 External links • Perlis, R. (2001), “Artin root numbers”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 22
Artin reciprocity law The Artin reciprocity law, established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory.[1] The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert’s product formula for the norm symbol. Artin’s result provided a partial solution to Hilbert’s ninth problem.
22.1 Significance Artin’s reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the Hasse local–global principle and the use of the Frobenius elements. Together with the Takagi existence theorem, it is used to describe the abelian extensions of K in terms of the arithmetic of K and to understand the behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of global class field theory. It can be used to prove that Artin L-functions are meromorphic and for the proof of the Chebotarev density theorem.[2] Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the transfer homomorphism of I. Schur and used the reciprocity law to translate the principalization problem for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups.[3]
22.2 Finite extensions of global fields The definition of the Artin map for a finite abelian extension L/K of global fields (such as a finite abelian extension of Q) has a concrete description in terms of prime ideals and Frobenius elements. If p is a prime of K then the decomposition groups of primes P above p are equal in Gal(L/K) since the latter group is abelian. If p is unramified in L, then the decomposition group Dp is canonically isomorphic to the Galois group of the extension of residue fields O(L,P /P ) over OK,p /p . There is therefore a canonically defined Frobenius element in Gal(L/K) denoted by Frobp or
L/K p
. If Δ denotes the relative discriminant of L/K, the Artin symbol (or Artin
∆ , by linearity: map, or (global) reciprocity map) of L/K is defined on the group of prime-to-Δ fractional ideals, IK
(
L/K ·
) :
∆ IK m ∏ pni i i=1
−→ 7→
Gal(L/K) )n m ( ∏ L/K i . pi i=1
The Artin reciprocity law (or global reciprocity law) states that there is a modulus c of K such that the Artin map induces an isomorphism ∼
c IK /i(Kc,1 )NL/K (ILc )−→Gal(L/K)
62
22.3. COHOMOLOGICAL INTERPRETATION
63
where K ,₁ is the ray modulo c, NL/K is the norm map associated to L/K and ILc is the fractional ideals of L prime to c. Such a modulus c is called a defining modulus for L/K. The smallest defining modulus is called the conductor of L/K and typically denoted f(L/K) .
22.2.1
Examples
Quadratic fields √
If d ̸= 1 is a squarefree integer, K = Q, and L=Q( d) , then the Galois group Gal(L/Q) can be identified with {±1}. The discriminant Δ of L over Q is d or 4d depending on whether d ≡ 1 (mod 4) or not. The Artin map is then defined on primes p that do not divide Δ by ( p 7→ where
) ∆ p ( ) ∆ p
is the Kronecker symbol.[4] More specifically, the conductor of L/Q is the principal ideal (Δ) or (Δ)∞
according to whether Δ is positive or negative,[5] and the Artin map on a prime-to-Δ ( ideal) (n) is given by the Kronecker ( ) ∆ symbol ∆ . This shows that a prime p is split or inert in L according to whether is 1 or −1. n p Cyclotomic fields Let m>1 be either an odd integer or a multiple of 4, let ζm be a primitive mth root of unity, and let L = Q(ζm) be the mth cyclotomic field. The Galois group Gal(L/Q) can be identified with (Z/mZ)× by sending σ to aσ given by the rule
aσ σ(ζm ) = ζm .
The conductor of L/Q is (m)∞,[6] and the Artin map on a prime-to-m ideal (n) is simply n (mod m) in (Z/mZ)× .[7]
22.2.2
Relation to quadratic reciprocity
Let p and ℓ be distinct odd primes. For convenience, let ℓ* = (−1)(ℓ−1)/2 ℓ (which is always 1 (mod 4)). Then, quadratic reciprocity states that (
ℓ∗ p
) =
(p) ℓ
. √
The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field F =Q( ℓ∗ ) and the cyclotomic field L=Q(ζℓ ) as follows.[4] First, F is a subfield of L, so if H = Gal(L/F) and G = Gal(L/Q), then Gal(F/Q) = G/H. Since the latter has order 2, the subgroup H must be the group of squares in (Z/ℓZ)× . A basic property of the Artin symbol says that for every prime-to-ℓ ideal (n) (
F /Q (n)
)
( =
L/Q (n)
) (mod H).
When n = p, this shows that
( ∗) ℓ p
= 1 if, and only if, p (mod ℓ) is in H, i.e. if, and only if, p is a square modulo ℓ.
22.3 Cohomological interpretation Let Lv⁄Kv be a Galois extension of local fields with Galois group G. The local reciprocity law describes a canonical isomorphism
64
CHAPTER 22. ARTIN RECIPROCITY LAW
ab θv : Kv× /NLv /Kv (L× v)→G ,
called the local Artin symbol, the local reciprocity map or the norm residue symbol.[8][9] Let L⁄K be a Galois extension of global fields and CL stand for the idèle class group of L. The maps θv for different places v of K can be assembled into a single global symbol map by multiplying the local components of an idèle class. One of the statements of the Artin reciprocity law is that this results in the canonical isomorphism[10][11]
θ : CK /NL/K (CL ) → Gal(L/K)ab . A cohomological proof of the global reciprocity law can be achieved by first establishing that
(Gal(K sep /K), lim CL ) −→ constitutes a class formation in the sense of Artin and Tate.[12] Then one proves that
ˆ 0 (Gal(L/K), CL ) ≃ H ˆ −2 (Gal(L/K), Z), H ˆ i denote the Tate cohomology groups. Working out the cohomology groups establishes that θ is an isomorwhere H phism.
22.4 Alternative statement An alternative version of the reciprocity law, leading to the Langlands program, connects Artin L-functions associated to abelian extensions of a number field with Hecke L-functions associated to characters of the idèle class group.[13] A Hecke character (or Größencharakter) of a number field K is defined to be a quasicharacter of the idèle class group of K. Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL(1) over the ring of adeles of K.[14] Let E⁄K be an abelian Galois extension with Galois group G. Then for any character σ: G → C× (i.e. one-dimensional complex representation of the group G), there exists a Hecke character χ of K such that
Hecke LArtin (χ, s) E/K (σ, s) = LK
where the left hand side is the Artin L-function associated to the extension with character σ and the right hand side is the Hecke L-function associated with χ, Section 7.D of.[14] The formulation of the Artin reciprocity law as an equality of L-functions allows formulation of a generalisation to n-dimensional representations, though a direct correspondence is still lacking.
22.5 Notes [1] Helmut Hasse, History of Class Field Theory, in Algebraic Number Theory, edited by Cassels and Frölich, Academic Press, 1967, pp. 266–279 [2] Jürgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, Chapter VII [3] Artin, Emil (December 1929), “Idealklassen in oberkörpern und allgemeines reziprozitätsgesetz”, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 7 (1): 46–51, doi:10.1007/BF02941159. [4] Lemmermeyer 2000, §3.2 [5] Milne 2008, example 3.11
22.6. REFERENCES
65
[6] Milne 2008, example 3.10 [7] Milne 2008, example 3.2 [8] Serre (1967) p.140 [9] Serre (1979) p.197 [10] Neukirch (1999) p.391 [11] Jürgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, p. 408. In fact, a more precise version of the reciprocity law keeps track of the ramification. [12] Serre (1979) p.164 [13] James Milne, Class Field Theory [14] Gelbart, Stephen S. (1975), Automorphic forms on adèle groups, Annals of Mathematics Studies 83, Princeton, N.J.: Princeton University Press, MR 0379375.
22.6 References • Emil Artin, Über eine neue Art von L-Reihen, Abh. Math. Semin. Univ. Hamburg, 3 (1924), 89–108; Collected Papers, Addison Wesley, 1965, 105–124 • Emil Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Semin. Univ. Hamburg, 5 (1927), 353–363; Collected Papers, 131–141 • Emil Artin, Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetzes, Abh. Math. Semin. Univ. Hamburg, 7 (1930), 46–51; Collected Papers, 159–164 • Frei, Günther (2004), “On the history of the Artin reciprocity law in abelian extensions of algebraic number fields: how Artin was led to his reciprocity law”, in Olav Arnfinn Laudal; Ragni Piene, The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3-−8, 2002, Berlin: Springer-Verlag, pp. 267–294, ISBN 978-3-540-43826-7, MR 2077576, Zbl 1065.11001 • Janusz, Gerald (1973), Algebraic Number Fields, Pure and Applied Mathematics 55, Academic Press, ISBN 0-12-380250-4 • Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics 110 (2 ed.), New York: SpringerVerlag, ISBN 978-0-387-94225-4, MR 1282723 • Lemmermeyer, Franz (2000), Reciprocity laws: From Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-66957-9, MR 1761696, Zbl 0949.11002 • Milne, James (2008), Class field theory (v4.0 ed.), retrieved 2010-02-22 • Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021 • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Translated from the French by Marvin Jay Greenberg, New York, Heidelberg, Berlin: Springer-Verlag, ISBN 3-540-90424-7, Zbl 0423.12016 • Serre, Jean-Pierre (1967), “VI. Local class field theory”, in Cassels, J.W.S.; Fröhlich, A., Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union, London: Academic Press, pp. 128–161, Zbl 0153.07403 • Tate, John (1967), “VII. Global class field theory”, in Cassels, J.W.S.; Fröhlich, A., Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union, London: Academic Press, pp. 162– 203, Zbl 0153.07403
Chapter 23
Artin transfer (group theory) In the mathematical field of group theory, an Artin transfer is a certain homomorphism from a group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin’s reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, the kernels and targets of Artin transfers have recently turned out to be compatible with parent-descendant relations between finite p-groups, which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These methods of pattern recognition are useful in purely group theoretic context, as well as for applications in algebraic number theory concerning higher p-class groups and Hilbert p-class field towers.
23.1 Transversals of a subgroup Let G be a group and H < G be a subgroup of finite index n = (G : H) . Definitions.
[1]
• A left transversal of H in G is an ordered system (g1 , . . . , gn ) of representatives for the left ∪n cosets of H in G such that G = ˙ i=1 gi H is a disjoint union. • Similarly, a right transversal of H in G is an ordered system (d1 , . . . , dn ) of representatives for ∪n the right cosets of H in G such that G = ˙ i=1 Hdi is a disjoint union. Remarks.
[2]
• For any transversal of H in G , there exists a unique subscript 1 ≤ i0 ≤ n such that gi0 ∈ H , resp. di0 ∈ H . Of course, this element may be, but need not be, replaced by the neutral element 1. • If G is non-abelian and H is not a normal subgroup of G , then we can only say that the inverse elements (g1−1 , . . . , gn−1 ) of a left transversal (g1 , . . . , gn ) form a right transversal of H in G , ∪n ∪n ∪n ∪n since G = ˙ i=1 gi H implies G = G−1 = ˙ i=1 (gi H)−1 = ˙ i=1 H −1 gi−1 = ˙ i=1 Hgi−1 . • However, if H ◁ G is a normal subgroup of G , then any left transversal is also a right transversal of H in G , since xH = Hx for each x ∈ G .
23.2 Permutation representation Suppose (g1 , . . . , gn ) is a left transversal of a subgroup H < G of finite index n = (G : H) in a group G . A fixed element x ∈ G gives rise to a unique permutation πx ∈ Sn of the left cosets of H in G such that xgi H = gπx (i) H , resp. xgi ∈ gπx (i) H , resp. hx (i) := gπ−1 xgi ∈ H , for each 1 ≤ i ≤ n . x (i) 66
23.3. ARTIN TRANSFER
67
Similarly, if (d1 , . . . , dn ) is a right transversal of H in G , then a fixed element x ∈ G gives rise to a unique permutation ρx ∈ Sn of the right cosets of H in G such that Hdi x = Hdρx (i) , resp. di x ∈ Hdρx (i) , resp. ηx (i) := di xd−1 ρx (i) ∈ H , for each 1 ≤ i ≤ n . Definition.
[1]
The mapping G → Sn , x 7→ πx , resp. x 7→ ρx , is called the permutation representation of G in Sn with respect to (g1 , . . . , gn ) , resp. (d1 , . . . , dn ) . The mapping G → H n × Sn , x 7→ (hx (1), . . . , hx (n); πx ) , resp. x 7→ (ηx (1), . . . , ηx (n); ρx ) , is called the monomial representation of G in H n × Sn with respect to (g1 , . . . , gn ) , resp. (d1 , . . . , dn ) . Remark. For the special right transversal (g1−1 , . . . , gn−1 ) associated to the left transversal (g1 , . . . , gn ) we have ηx (i) = gi−1 xgρx (i) but on the other hand hx (i)−1 = (gπ−1 xgi )−1 = gi−1 x−1 gπx (i) = gi−1 x−1 gρx−1 (i) = ηx−1 (i) x (i) , for each 1 ≤ i ≤ n . This relation simultaneously shows that, for any x ∈ G , the associated permutation representations are connected by ρx−1 = πx , and the associated monomial representations are connected additionally by ηx−1 (i) = hx (i)−1 , for each 1 ≤ i ≤ n .
23.3 Artin transfer Let G be a group and H < G be a subgroup of finite index n = (G : H) . Assume that (g1 , . . . , gn ) , resp. (d1 , . . . , dn ) , is a left, resp. right, transversal of H in G . Definition.
[2] [3]
Then the Artin transfer TG,H : G → H/H ′ from G to the abelianization of H with respect to (g1 , . . . , gn ) , ∏n ∏n (g) resp. (d1 , . . . , dn ) , is defined by TG,H (x) := i=1 gπ−1 xgi · H ′ or briefly TG,H (x) = i=1 hx (i) · H ′ , resp. x (i) ∏n ∏n (d) ′ ′ TG,H (x) := i=1 di xd−1 i=1 ηx (i) · H , for x ∈ G . ρx (i) · H or briefly TG,H (x) =
23.3.1
Independence of the transversal n
Assume that (γ1 , . . . , γn ) is another left transversal of H in G such that G = ∪˙ i=1 γi H . Then there exists a unique permutation σ ∈ Sn such that gi H = γσ(i) H , for all 1 ≤ i ≤ n . Consequently, hi := gi−1 γσ(i) ∈ H , resp. γσ(i) = gi hi with hi ∈ H , for all 1 ≤ i ≤ n . For a fixed element x ∈ G , there exists a unique permutation λx ∈ Sn such that we have γλx (σ(i)) H = xγσ(i) H = xgi hi H = xgi H = gπx (i) H = gπx (i) hπx (i) H = γσ(πx (i)) H , for all 1 ≤ i ≤ n . Therefore, the permutation representation of G with respect to (γ1 , . . . , γn ) is given by λx ◦ σ = σ ◦ πx , resp. λx = σ ◦ πx ◦ σ −1 ∈ Sn , for x ∈ G . Furthermore, for the connection between the elements kx (i) := γλ−1 xγi ∈ H and hx (i) := gπ−1 xgi ∈ H , we obtain kx (σ(i)) = γλ−1 xγσ(i) = x (i) x (i) x (σ(i)) −1 −1 −1 −1 −1 γσ(πx (i)) xgi hi = (gπx (i) hπx (i) ) xgi hi = hπx (i) gπx (i) xgi hi = hπx (i) hx (i)hi , for all 1 ≤ i ≤ n . Finally, due to the commutativity of the quotient group H/H ′ and the fact that σ, πx are permutations, the Artin transfer ∏n ∏n (γ) ′ turns out to be independent of the left transversal: TG,H (x) = i=1 kx (σ(i)) · H ′ = i=1 h−1 πx (i) hx (i)hi · H ∏n ∏n ∏ ∏ ∏ (g) n n n ′ ′ ′ = i=1 hx (i) i=1 h−1 i=1 hi · H = i=1 hx (i) · 1 · H = i=1 hx (i) · H = TG,H (x) , as defined above. πx (i) It is clear that a similar proof shows that the Artin transfer is independent of the choice between two different right transversals. It remains to show that the Artin transfer with respect to a right transversal coincides with the Artin transfer with respect to a left transversal. For this purpose, we select the special right transversal (g1−1 , . . . , gn−1 ) associated to the left transversal (g1 , . . . , gn ) . Using the commutativity of H/H ′ and the remark in the previous ∏n ∏n ∏n (g −1 ) section, we consider the expression TG,H (x) = i=1 gi−1 xgρx (i) · H ′ = i=1 ηx (i) · H ′ = i=1 hx−1 (i)−1 · ∏n (g) (g) H ′ = ( i=1 hx−1 (i) · H ′ )−1 = (TG,H (x−1 ))−1 = TG,H (x) . The last step is justified by the fact that the Artin transfer is a homomorphism. This will be shown in the following section.
23.3.2
Homomorphisms
∏n ∏n Let x, y ∈ G be two elements with transfer images TG,H (x) = i=1 gπ−1 xgi ·H ′ and TG,H (y) = j=1 gπ−1 ygj · x (i) y (j) H ′ . Since H/H ′ is abelian and πy is a permutation, we can change the order of the factors in the following product:
68
CHAPTER 23. ARTIN TRANSFER (GROUP THEORY)
∏n ∏n ∏n ∏n xgi H ′ · j=1 gπ−1 ygj ·H ′ = j=1 gπ−1 xgπy (j) H ′ · j=1 gπ−1 ygj ·H ′ TG,H (x)·TG,H (y) = i=1 gπ−1 x (i) y (j) x (πy (j)) y (j) ∏n ∏ n −1 = j=1 gπ−1 xgπy (j) gπ−1 ygj · H ′ = j=1 g(π xygj · H ′ = TG,H (xy) . This relation simultanex (πy (j)) y (j) x ◦πy )(j)) ously shows that the Artin transfer TG,H and the permutation representation G → Sn , x 7→ πx are homomorphisms, since πxy = πx ◦ πy . It is illuminating to restate the homomorphism property of∏the Artin transfer in terms of the monomial representation. ∏n n The images of the factors x, y are given by TG,H (x) = i=1 hx (i) · H ′ and TG,H (y) = j=1 hy (j) · H ′ . The ∏n image of the product xy turned out to be TG,H (xy) = j=1 hx (πy (j)) · hy (j) · H ′ , which is a very peculiar law of composition discussed in more detail in the following section. The law reminds of the crossed homomorphisms x 7→ hx in the first cohomology group H1 (G, M ) of a G -module M , which have the property hxy = hyx · hy .
23.3.3
Wreath product of H and S(n)
The peculiar structures which arose in the previous section can also be interpreted by endowing the cartesian product H n × Sn with a special law of composition known as the wreath product H ≀ Sn of the groups H and Sn with respect to the set {1, . . . , n} . For x, y ∈ G , it is given by (hx (1), . . . , hx (n); πx ) · (hy (1), . . . , hy (n); πy ) := (hx (πy (1)) · hy (1), . . . , hx (πy (n)) · hy (n); πx ◦ πy ) = (hxy (1), . . . , hxy (n); πxy ) , which causes the monomial representation G → H ≀ Sn , x 7→ (hx (1), . . . , hx (n); πx ) also to be a homomorphism. In fact, it is an injective homomorphism, also called a monomorphism or embedding, in contrast to the permutation representation.
23.3.4
Composition
Let G be a group with nested subgroups K ≤ H ≤ G such that the index (G : K) = (G : H) · (H : K) = n · m is finite. Then the Artin transfer TG,K is the compositum of the induced transfer T˜H,K : H/H ′ → K/K ′ and the Artin transfer TG,H , that is, TG,K = T˜H,K ◦ TG,H . This can be seen in the following manner. n If (g1 , . . . , gn ) is a left transversal of H in G and (h1 , . . . , hm ) is a left transversal of K in H , that is G = ∪˙ i=1 gi H m n m and H = ∪˙ j=1 hj K , then G = ∪˙ i=1 ∪˙ j=1 gi hj K is a disjoint left coset decomposition of G with respect to K . Given two elements x ∈ G and y ∈ H , there exist unique permutations πx ∈ Sn , and σy ∈ Sm , such that hx (i) := gπ−1 xgi ∈ H , for each 1 ≤ i ≤ n , and ky (j) := h−1 σy (j) yhj ∈ K , for each 1 ≤ j ≤ m . Then x (i) ∏n ∏ m TG,H (x) = hx (i) · H ′ , and T˜H,K (y · H ′ ) = TH,K (y) = ky (j) · K ′ . For each pair of subscripts i=1
j=1
1 ≤ i ≤ n and 1 ≤ j ≤ m , we have xgi hj = gπx (i) gπ−1 xgi hj = gπx (i) hx (i)hj = gπx (i) hσyi (j) kyi (j) , resp. x (i) −1 −1 hσy (j) gπx (i) xgi hj = kyi (j) , where yi := hx (i) . Therefore, the image of x under the Artin transfer TG,K is given i ∏n ∏m ∏n ∏m ∏n ∏m −1 −1 ′ by TG,K (x) = i=1 j=1 kyi (j) · K ′ = i=1 j=1 h−1 i=1 j=1 hσyi (j) hx (i)hj · σyi (j) gπx (i) xgi hj · K = ∏ ∏ ∏ ∏ ∏ n n n m n hx (i) · H ′ ) = yi · H ′ ) = T˜H,K ( K′ = h−1 yi hj · K ′ = T˜H,K (yi · H ′ ) = T˜H,K ( i=1
j=1
T˜H,K (TG,H (x)) .
σyi (j)
i=1
i=1
i=1
Finally, we want to emphasize the structural peculiarity of the corresponding monomial representation G → K n·m × Sn·m , x 7→ (ℓx (1, 1), . . . , ℓx (n, m); γx ) , defining ℓx (i, j) := ((gh)γx (i,j) )−1 x(gh)(i,j) ∈ K for a permutation γx ∈ Sn·m , and using the symbolic notation (gh)(i,j) := gi hj for all pairs of subscripts 1 ≤ i ≤ n , 1 ≤ j ≤ m . −1 The preceding proof has shown that ℓx (i, j) = h−1 σyi (j) gπx (i) xgi hj . Therefore, the action of the permutation γx on the set [1, n] × [1, m] is given by γx (i, j) = (πx (i), σhx (i) (j)) . The action on the second component depends on the first component (via the permutation σhx (i) ∈ Sm to be selected) whereas the action on the first component is independent of the second component. Therefore, the permutation γx ∈ Sn·m can be identified with the multiplet
n (πx ; σhx (1) , . . . , σhx (n) ) ∈ Sn × Sm ,
which will be written in twisted form in the next section.
23.3.5
Wreath product of S(m) and S(n)
The permutations γx , which arose as second components of the monomial representation G → K n·m × Sn·m , x 7→ (ℓx (1, 1), . . . , ℓx (n, m); γx ) , in the previous section, are of a very special kind. They belong to the stabilizer
23.4. COMPUTATIONAL IMPLEMENTATION
69
of the natural equipartition of the set [1, n] × [1, m] into the n rows of the corresponding matrix (rectangular array). Using the peculiarities of the composition of Artin transfers in the previous section, we show that this stabilizer is isomorphic to the wreath product Sm ≀ Sn of the groups Sm and Sn with respect to the set {1, . . . , n} , whose n underlying set Sm × Sn is endowed with the following law of composition γx · γz = (σhx (1) , . . . , σhx (n) ; πx ) · (σhz (1) , . . . , σhz (n) ; πz ) = (σhx (πz (1)) ◦ σhz (1) , . . . , σhx (πz (n)) ◦ σhz (n) ; πx ◦ πz ) = (σhxz (1) , . . . , σhxz (n) ; πxz ) = γxz for all x, z ∈ G . This law reminds of the chain rule D(g ◦ f )(x) = D(g)(f (x)) ◦ D(f )(x) for the Fréchet derivative in x ∈ E of the compositum of differentiable functions f : E → F and g : F → G between normed spaces. The above considerations establish a third representation, the stabilizer representation, G → Sm ≀Sn , x 7→ (σhx (1) , . . . , σhx (n) ; πx ) of the group G in the wreath product Sm ≀ Sn , similar to the permutation representation and the monomial representation. As opposed to the latter, the stabilizer representation cannot be injective, in general. For instance, if G is infinite.
23.3.6
Cycle decomposition
Let (g1 , . . . , gn ) be a left transversal of a subgroup H < G of finite index n = (G : H) in a group G . Suppose the element x ∈ G gives rise to the permutation πx ∈ Sn of the left cosets of H in G such that xgi H = gπx (i) H , resp. gπ−1 xgi =: hx (i) ∈ H , for each 1 ≤ i ≤ n . x (i) ∏t If πx has the decomposition πx = j=1 ζj into pairwise disjoint cycles ζj ∈ Sn of lengths fj ≥ 1 , which is unique up to the ordering of the cycles, more explicitly, if (gj H, gζj (j) H, gζj2 (j) H, . . . , g fj −1 H) = (gj H, xgj H, x2 gj H, . . . , xfj −1 gj H) ζj (j) ∑t , for 1 ≤ j ≤ t , and j=1 fj = n , then the image of x under the Artin transfer TG,H is given by TG,H (x) = ∏t −1 fj ′ j=1 gj x gj · H . The reason for this fact is that we obtain another left transversal of H in G by putting γj,k := xk gj for 0 ≤ k ≤ fj −1 t fj −1 k and 1 ≤ j ≤ t , since G = ∪˙ j=1 ∪˙ k=0 x gj H . Let us fix a value of 1 ≤ j ≤ t . For 0 ≤ k ≤ fj − 2 , we k k+1 have xγj,k = xx gj = x gj = γj,k+1 = γj,k+1 · 1 , resp. hx (j, k) = 1 . However, for k = fj − 1 , we obtain xγj,fj −1 = xxfj −1 gj = xfj gj ∈ gj H , resp. gj−1 xfj gj = hx (j, fj − 1) ∈ H . Consequently, ∏t ∏fj −2 ∏t ∏t ∏fj −1 1) · hx (j, fj − 1) · H ′ = j=1 gj−1 xfj gj · H ′ . The hx (j, k) · H ′ = j=1 ( k=0 TG,H (x) = j=1 k=0 t cycle decomposition corresponds to a double coset decomposition G = ∪˙ j=1 ⟨x⟩gj H of the group G modulo the cyclic group ⟨x⟩ and the subgroup H . It was actually this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper.[3]
23.3.7
Normal subgroup
Let H ◁ G be a normal subgroup of finite index n = (G : H) in a group G . Then we have xH = Hx , for all x ∈ G , and there exists the quotient group G/H of order n . For an element x ∈ G , we let f := ord(xH) denote the order of the coset xH in G/H . Then, ⟨xH⟩ is a cyclic subgroup of order f of G/H , and a (left) transversal t (g1 , . . . , gt ) of the subgroup ⟨x, H⟩ in G , where t = n/f and G = ∪˙ j=1 gj ⟨x, H⟩ , can be extended to a (left) t f −1 transversal G = ∪˙ j=1 ∪˙ k=0 gj xk H of H in G . Hence, the formula for the image of x under the Artin transfer TG,H ∏t in the previous section takes the particular shape TG,H (x) = j=1 gj−1 xf gj · H ′ with exponent f independent of j. In particular, the inner transfer of an element x ∈ H of order f = 1 is given as a symbolic power TG,H (x) = ∑t ∏t ∏t ∑t gj −1 ′ gj ′ j=1 · H ′ with the trace element TrG (H) = j=1 gj ∈ Z[G] of H j=1 gj xgj · H = j=1 x · H = x in G as symbolic exponent. The other extreme is the outer transfer of an element x ∈ G \ H which generates G ∏1 modulo H , that is G = ⟨x, H⟩ and f = n . It is simply an n th power TG,H (x) = j=1 1−1 · xn · 1 · H ′ = xn · H ′ . Explicit implementations of Artin transfers in the simplest situations are presented in the following section.
23.4 Computational implementation
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23.4.1
CHAPTER 23. ARTIN TRANSFER (GROUP THEORY)
Abelianization of type (p,p)
Let G be a p-group with abelianization G/G′ of elementary abelian type (p, p) . Then G has p+1 maximal subgroups Hi < G (1 ≤ i ≤ p + 1) of index (G : Hi ) = p . For each 1 ≤ i ≤ p + 1 , let Ti : G → Hi /Hi′ be the Artin transfer homomorphism from G to the abelianization of Hi . According to Burnside’s basis theorem, the group G has generator rank d(G) = 2 and can therefore be generated as G = ⟨x, y⟩ by two elements x, y such that xp , y p ∈ G′ . For each of the normal subgroups Hi ◁ G , a generator hi with respect to G′ , and a generator ti of a transversal must be given such that Hi = ⟨hi , G′ ⟩ and G = ⟨ti , Hi ⟩ . A convenient selection is given by h1 = y , t1 = x , and hi = xy i−2 , ti = y , for all 2 ≤ i ≤ p + 1 . Then, for each 1 ≤ i ≤ p + 1 , it is sufficient to define the 1+t +t2 +···+tp−1
Tr (H )
i inner transfer by Ti (hi ) = hi G i · Hi′ = hi i i · Hi′ , which can also be expressed as a product −p+1 p−1 −1 −2 2 ′ p p ′ of two pth powers hi · ti hi ti · ti hi ti · · · ti hi ti · Hi = (hi t−1 i ) ti · Hi , and the outer transfer as a p ′ complete pth power by Ti (ti ) = ti · Hi . The reason is that f = ord(hi Hi ) = 1 and f = ord(ti Hi ) = p . It should be pointed out that the complete specification of the Artin transfers also requires explicit knowledge of the derived subgroups Hi′ . Since G′ is a normal subgroup of index p in Hi , a certain general reduction is possible by Hi′ = [Hi , Hi ] = [G′ , Hi ] = (G′ )hi −1 , [4] but a presentation of G must be known for determining generators of G′ = ⟨s1 , . . . , sn ⟩ , whence Hi′ = ⟨[s1 , hi ], . . . , [sn , hi ]⟩ .
23.4.2
Abelianization of type (p2 ,p)
Let G be a p-group with abelianization G/G′ of non-elementary abelian type (p2 , p) . Then G has p + 1 maximal subgroups Hi < G (1 ≤ i ≤ p + 1) of index (G : Hi ) = p and p + 1 subgroups Ui < G (1 ≤ i ≤ p + 1) of index (G : Ui ) = p2 . For each 1 ≤ i ≤ p + 1 , let T1,i : G → Hi /Hi′ , resp. T2,i : G → Ui /Ui′ , be the Artin transfer homomorphism from G to the abelianization of Hi , resp. Ui . Burnside’s basis theorem asserts that the group G has generator rank d(G) = 2 and can therefore be generated as G = ⟨x, y⟩ by two elements x, y 2 such that xp , y p ∈ G′ . We begin by considering the first layer of subgroups. For each of the normal subgroups Hi ◁ G (1 ≤ i ≤ p) , we select a generator hi = xy i−1 such that Hi = ⟨hi , G′ ⟩ . These are the cases where the factor group Hi /G′ is cyclic of order p2 . However, for the distinguished maximal subgroup Hp+1 , for which the factor group Hp+1 /G′ is bicyclic of type (p, p) , we need two generators hp+1 = y and h0 = xp such that Hp+1 = ⟨hp+1 , h0 , G′ ⟩ . Further, a generator ti of a transversal must be given such that G = ⟨ti , Hi ⟩ , for each 1 ≤ i ≤ p + 1 . It is convenient to define ti = y , for 1 ≤ i ≤ p , and tp+1 = x . Then, for each 1 ≤ i ≤ p + 1 1+t +t2 +...+tp−1
p p ′ i , we have the inner transfer T1,i (hi ) = hi G i · Hi′ = hi i i · Hi′ , which equals (hi t−1 i ) ti · Hi , p ′ and the outer transfer T1,i (ti ) = ti · Hi , since f = ord(hi Hi ) = 1 and f = ord(ti Hi ) = p . Now we continue by considering the second layer of subgroups. For each of the normal subgroups Ui ◁ G (1 ≤ i ≤ p + 1) , we select a generator u1 = y , ui = xp y i−1 for 2 ≤ i ≤ p , and up+1 = xp , such that Ui = ⟨ui , G′ ⟩ . Among these subgroups, the Frattini subgroup Up+1 = ⟨xp , G′ ⟩ = Gp · G′ is particularly distinguished. A uniform way of defining generators ti , wi of a transversal such that G = ⟨ti , wi , Ui ⟩ , is to set ti = x, wi = xp , for 1 ≤ i ≤ p , and tp+1 = x, wp+1 = y . Since f = ord(ui Ui ) = 1 , but on the other hand f = ord(ti Ui ) = p2 and f = ord(wi Ui ) = p , for 1 ≤ i ≤ p + 1 , with the single exception that f = ord(tp+1 Up+1 ) = p , we obtain the following expressions ∑p−1 ∑p−1 j k ∏p−1 ∏p−1 w t Tr (U ) for the inner transfer T2,i (ui ) = ui G i · Ui′ = ui j=0 k=0 i i · Ui′ = j=0 k=0 (wij tki )−1 ui wij tki · Ui′ , Tr (H )
2
2
p−1
′ and for the outer transfer T2,i (ti ) = tpi · Ui′ , exceptionally T2,p+1 (tp+1 ) = (tpp+1 )1+wp+1 +wp+1 +...+wp+1 · Up+1 p−1
, and T2,i (wi ) = (wip )1+ti +ti +...+ti · Ui′ , for 1 ≤ i ≤ p + 1 . Again, it should be emphasized that the structure of the derived subgroups Hi′ and Ui′ must be known to specify the action of the Artin transfers completely. 2
23.5 Transfer kernels and targets Let G be a group with finite abelianization G/G′ . Suppose that (Hi )i∈I denotes the family of all subgroups Hi ◁ G which contain the commutator subgroup G′ and are therefore necessarily normal, enumerated by means of the finite index set I . For each i ∈ I , let Ti := TG,Hi be the Artin transfer from G to the abelianization Hi /Hi′ . Definition.
[5]
The family of normal subgroups κH (G) = (ker(Ti ))i∈I is called the transfer kernel type (TKT) of G with respect to (Hi )i∈I , and the family of abelianizations (resp. their abelian type invariants) τH (G) = (Hi /Hi′ )i∈I is called the transfer target type (TTT) of G with respect to (Hi )i∈I . Both families are also called multiplets whereas a single component will be referred to as a singulet.
23.6. ABELIANIZATION OF TYPE (P,P)
71
Important examples for these concepts are provided in the following two sections.
23.6 Abelianization of type (p,p) Let G be a p-group with abelianization G/G′ of elementary abelian type (p, p) . Then G has p+1 maximal subgroups Hi < G (1 ≤ i ≤ p + 1) of index (G : Hi ) = p . For each 1 ≤ i ≤ p + 1 , let Ti : G → Hi /Hi′ be the Artin transfer homomorphism from G to the abelianization of Hi . Definition. The family of normal subgroups κH (G) = (ker(Ti ))1≤i≤p+1 is called the transfer kernel type (TKT) of G with respect to H1 , . . . , Hp+1 . Remarks. • For brevity, the TKT{is identified with the multiplet (κ(i))1≤i≤p+1 , whose integer components 0 if ker(Ti ) = G, are given by κ(i) = Here, we take into considj if ker(Ti ) = Hj some for 1 ≤ j ≤ p + 1. eration that each transfer kernel ker(Ti ) must contain the commutator subgroup G′ of G , since the transfer target Hi /Hi′ is abelian. However, the minimal case ker(Ti ) = G′ cannot occur. • A renumeration of the maximal subgroups Ki = Hπ(i) and of the transfers Vi = Tπ(i) by means of a permutation π ∈ Sp+1 gives rise to a new TKT λK (G) ={(ker(Vi ))1≤i≤p+1 with respect to 0 if ker(Vi ) = G, K1 , . . . , Kp+1 , identified with (λ(i))1≤i≤p+1 , where λ(i) = j if ker(Vi ) = Kj some for 1 ≤ j ≤ p + 1. It is adequate to view the TKTs λK (G) ∼ κH (G) as equivalent. Since we have Kλ(i) = ker(Vi ) = ker(Tπ(i) ) = Hκ(π(i)) = Kπ˜ −1 (κ(π(i))) , the relation between λ and κ is given by λ = π ˜ −1 ◦ κ ◦ π . Therefore, λ is another representative of the orbit κ Sp+1 of κ under the operation (π, µ) 7→ π ˜ −1 ◦ µ ◦ π of the symmetric group Sp+1 on the set of all mappings from {1, . . . , p + 1} to {0, . . . , p + 1} , where the extension π ˜ ∈ Sp+2 of the permutation π ∈ Sp+1 is defined by π ˜ (0) = 0 , and formally H0 = G , K0 = G . Definition. The orbit κ(G) = κ Sp+1 of any representative κ is an invariant of the p-group G and is called its transfer kernel type, briefly TKT. Remark. Let #H0 (G) := #{1 ≤ i ≤ p + 1 | κ(i) = 0} denote the counter of total transfer kernels ker(Ti ) = G , which is an invariant of the group G . In 1980, S. M. Chang and R. Foote [6] proved that, for any odd prime p and for any integer 0 ≤ n ≤ p + 1 , there exist metabelian p-groups G having abelianization G/G′ of type (p, p) such that #H0 (G) = n . However, for p = 2 , there do not exist non-abelian 2 -groups G with G/G′ ≃ (2, 2) , which must be metabelian of maximal class, such that #H0 (G) ≥ 2 . Only the elementary abelian 2 -group G = C2 × C2 has #H0 (G) = 3 . See Figure 5. In the following concrete examples for the counters #H0 (G) , and also in the remainder of this article, we use identifiers of finite p-groups in the SmallGroups Library by H. U. Besche, B. Eick and E. A. O'Brien .[7] [8] For p = 3 , we have • #H0 (G) = 0 for the extra special group G = ⟨27, 4⟩ of exponent 9 with TKT κ = (1111) (Figure 6), • #H0 (G) = 1 for the two groups G ∈ {⟨243, 6⟩, ⟨243, 8⟩} with TKTs κ ∈ {(0122), (2034)} (Figures 8 and 9), • #H0 (G) = 2 for the group G = ⟨243, 3⟩ with TKT κ = (0043) (Figure 4 in the article on descendant trees), • #H0 (G) = 3 for the group G = ⟨81, 7⟩ with TKT κ = (2000) (Figure 6), • #H0 (G) = 4 for the extra special group G = ⟨27, 3⟩ of exponent 3 with TKT κ = (0000) (Figure 6).
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23.7 Abelianization of type (p2 ,p) Let G be a p-group with abelianization G/G′ of non-elementary abelian type (p2 , p) . Then G possesses p + 1 maximal subgroups Hi < G (1 ≤ i ≤ p + 1) of index (G : Hi ) = p , and p + 1 subgroups Ui < G (1 ≤ i ≤ p + 1) of index (G : Ui ) = p2 . Assumption.
∏p+1 Suppose that Hp+1 = j=1 Uj is the distinguished maximal subgroup which is the product of all subgroups of 2 index p2 , and Up+1 = ∩p+1 j=1 Hj is the distinguished subgroup of index p which is the intersection of all maximal subgroups, that is the Frattini subgroup Φ(G) of G .
23.7.1
First layer
For each 1 ≤ i ≤ p + 1 , let T1,i : G → Hi /Hi′ be the Artin transfer homomorphism from G to the abelianization of Hi . Definition.
The family κ1,H,U (G) = (ker(T1,i ))1≤i≤p+1 is called the first layer transfer kernel type { of G with respect to 0 if ker(T1,i ) = Hp+1 , H1 , . . . , Hp+1 and U1 , . . . , Up+1 , and is identified with (κ1 (i))1≤i≤p+1 , where κ1 (i) = j if ker(T1,i ) = Uj some for 1 ≤ j ≤ p Remark. Here, we observe that each first layer transfer kernel is of exponent p with respect to G′ and consequently cannot coincide with Hj for any 1 ≤ j ≤ p , since Hj /G′ is cyclic of order p2 , whereas Hp+1 /G′ is bicyclic of type (p, p) .
23.7.2
Second layer
For each 1 ≤ i ≤ p + 1 , let T2,i : G → Ui /Ui′ be the Artin transfer homomorphism from G to the abelianization of Ui . Definition.
The family κ2,U,H (G) = (ker(T2,i ))1≤i≤p+1 is called the second layer transfer kernel { type of G with respect to 0 if ker(T2,i ) = G, U1 , . . . , Up+1 and H1 , . . . , Hp+1 , and is identified with (κ2 (i))1≤i≤p+1 , where κ2 (i) = j if ker(T2,i ) = Hj some for 1 ≤ j ≤ p
23.7.3
Transfer kernel type
Combining the information on the two layers, we obtain the (complete) transfer kernel type κH,U (G) = (κ1,H,U (G); κ2,U,H (G)) of the p-group G with respect to H1 , . . . , Hp+1 and U1 , . . . , Up+1 . Remark. The distinguished subgroups Hp+1 and Up+1 = Φ(G) are unique invariants of G and should not be renumerated. However, independent renumerations of the remaining maximal subgroups Ki = Hτ (i) (1 ≤ i ≤ p) and the transfers V1,i = T1,τ (i) by means of a permutation τ ∈ Sp , and of the remaining subgroups Wi = Uσ(i) (1 ≤ i ≤ p) of index p2 and the transfers V2,i = T2,σ(i) by means of a permutation σ ∈ Sp , give rise to new TKTs λ1,K,W (G) = (ker(V1,i )) {1≤i≤p+1 with respect to K1 , . . . , Kp+1 and W1 , . . . , Wp+1 , identi0 if ker(V1,i ) = Kp+1 , fied with (λ1 (i))1≤i≤p+1 , where λ1 (i) = and λ2,W,K (G) = j if ker(V1,i ) = Wj some for 1 ≤ j ≤ p + 1, (ker(V2,i{ ))1≤i≤p+1 with respect to W1 , . . . , Wp+1 and K1 , . . . , Kp+1 , identified with (λ2 (i))1≤i≤p+1 , where 0 if ker(V2,i ) = G, λ2 (i) = It is adequate to view the TKTs λ1,K,W (G) ∼ κ1,H,U (G) j if ker(V2,i ) = Kj some for 1 ≤ j ≤ p + 1. and λ2,W,K (G) ∼ κ2,U,H (G) as equivalent. Since we have Wλ1 (i) = ker(V1,i ) = ker(T1,ˆτ (i) ) = Uκ1 (ˆτ (i)) =
23.8. INHERITANCE FROM QUOTIENTS
73
Wσ˜ −1 (κ1 (ˆτ (i))) , resp. Kλ2 (i) = ker(V2,i ) = ker(T2,ˆσ(i) ) = Hκ2 (ˆσ(i)) = Kτ˜−1 (κ2 (ˆσ(i))) , the relations between λ1 and κ1 , resp. λ2 and κ2 , are given by λ1 = σ ˜ −1 ◦ κ1 ◦ τˆ , resp. λ2 = τ˜−1 ◦ κ2 ◦ σ ˆ . Therefore, Sp ×Sp λ = (λ1 , λ2 ) is another representative of the orbit κ of κ = (κ1 , κ2 ) under the operation ((σ, τ ), (µ1 , µ2 )) 7→ (˜ σ −1 ◦ µ1 ◦ τˆ, τ˜−1 ◦ µ2 ◦ σ ˆ ) of the product of two symmetric groups Sp × Sp on the set of all pairs of mappings from {1, . . . , p + 1} to {0, . . . , p + 1} , where the extensions π ˆ ∈ Sp+1 and π ˜ ∈ Sp+2 of a permutation π ∈ Sp are defined by π ˆ (p + 1) = π ˜ (p + 1) = p + 1 and π ˜ (0) = 0 , and formally H0 = K0 = G , Kp+1 = Hp+1 , U0 = W0 = Hp+1 , and Wp+1 = Up+1 = Φ(G) . Definition. The orbit κ(G) = κ Sp ×Sp of any representative κ = (κ1 , κ2 ) is an invariant of the p-group G and is called its transfer kernel type, briefly TKT.
23.7.4
Connections between layers
The Artin transfer T2,i : G → Ui /Ui′ from G to a subgroup Ui of index (G : Ui ) = p2 ( 1 ≤ i ≤ p + 1 ) is the compositum T2,i = T˜Hj ,Ui ◦ T1,j of the induced transfer T˜Hj ,Ui : Hj /Hj′ → Ui /Ui′ from Hj to Ui and the Artin transfer T1,j : G → Hj /Hj′ from G to Hj , for any intermediate subgroup Ui < Hj < G of index (G : Hj ) = p ( 1 ≤ j ≤ p + 1 ). There occur two situations: • For the subgroups U1 , . . . , Up only the distinguished maximal subgroup Hp+1 is an intermediate subgroup. • For the Frattini subgroup Up+1 = Φ(G) all maximal subgroups H1 , . . . , Hp+1 are intermediate subgroups. This causes restrictions for the transfer kernel type κ2 (G) of the second layer, since ker(T2,i ) = ker(T˜Hj ,Ui ◦T1,j ) ⊃ ker(T1,j ) , and thus • ker(T2,i ) ⊃ ker(T1,p+1 ) , for all 1 ≤ i ≤ p , p+1 • but even ker(T2,p+1 ) ⊃ ⟨∪j=1 ker(T1,j )⟩ .
Furthermore, when G = ⟨x, y⟩ with xp ∈ / G′ and y p ∈ G′ , an element xy k−1 ( 1 ≤ k ≤ p ) which is of order p2 ′ with respect to G , can belong to the transfer kernel ker(T2,i ) only if its p th power xp is contained in ker(T1,j ) , for all intermediate subgroups Ui < Hj < G , and thus: • xy k−1 ∈ ker(T2,i ) , for certain 1 ≤ i, k ≤ p , enforces the first layer TKT singulet κ1 (p + 1) = p+1, • but xy k−1 ∈ ker(T2,p+1 ) , for some 1 ≤ k ≤ p , even specifies the complete first layer TKT multiplet κ1 = ((p + 1)p+1 ) , that is κ1 (j) = p + 1 , for all 1 ≤ j ≤ p + 1 .
23.8 Inheritance from quotients The common feature of all parent-descendant relations between finite p-groups is that the parent π(G) is a quotient G/N of the descendant G by a suitable normal subgroup N ◁ G . Thus, an equivalent definition can be given by ˜ whose kernel ker(φ) plays the role of the normal subgroup N ◁ G selecting an epimorphism φ from G onto a group G . In the following sections, this point of view will be taken, generally for arbitrary groups.
23.8.1
Passing through the abelianization
If φ : G → A is a homomorphism from a group G to an abelian group A , then there exists a unique homomorphism φ˜ : G/G′ → A such that φ = φ˜ ◦ ω , where ω : G → G/G′ denotes the canonical projection. The kernel of φ˜ is given by ker(φ) ˜ = ker(φ)/G′ . The situation is visualized in Figure 1. The uniqueness of φ˜ is a consequence of the condition φ = φ˜ ◦ ω , which implies that φ˜ must be defined by ′ ′ ′ φ(xG ˜ ) = φ(ω(x)) ˜ = (φ˜ ◦ ω)(x) = φ(x) , for any x ∈ G . The relation φ(xG ˜ · yG′ ) = φ((xy)G ˜ ) = φ(xy) =
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CHAPTER 23. ARTIN TRANSFER (GROUP THEORY)
Figure 1: Factoring through the abelianization.
′ ′ φ(x) · φ(y) = φ(xG ˜ ) · φ(xG ˜ ) , for x, y ∈ G , shows that φ˜ is a homomorphism. For the commutator of x, y ∈ G , we have φ([x, y]) = [φ(x), φ(y)] = 1 , since A is abelian. Thus, the commutator subgroup G′ of G is contained in the kernel ker(φ) , and this finally shows that the definition of φ˜ is independent of the coset representative, xG′ = yG′ ′ −1 ′ ′ ′ ⇒ x−1 y ∈ G′ ≤ ker(φ) ⇒ φ(xG ˜ ) · φ(yG ˜ ) = φ(x ˜ −1 yG′ ) = φ(x−1 y) = 1 ⇒ φ(xG ˜ ) = φ(yG ˜ ).
23.8.2
TTT singulets
˜ be groups such that G ˜ = φ(G) is the image of G under an epimorphism φ : G → G ˜ and H ˜ = φ(H) Let G and G is the image of a subgroup H ≤ G .
23.8. INHERITANCE FROM QUOTIENTS
75
Figure 2: Epimorphisms and derived quotients.
˜ is the image of the commutator subgroup of H , that is H ˜ ′ = φ(H ′ ) . If ker(φ) ≤ H The commutator subgroup of H ′ ′ ˜ ˜ ˜ ˜ H ˜ ′ is epimorphic , then H ≃ H/ ker(φ) , φ induces a unique epimorphism φ˜ : H/H → H/H , and thus H/ ′ ′ ′ ′ ′ ˜ image of H/H , that is a quotient of H/H . Moreover, if even ker(φ) ≤ H , then H ≃ H / ker(φ) , the map φ˜ ˜ H ˜ ′ ≃ H/H ′ . See Figure 2 for a visualization of this scenario. is an isomorphism, and H/
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CHAPTER 23. ARTIN TRANSFER (GROUP THEORY)
The statements can be seen in the following manner. The image of the commutator subgroup is φ(H ′ ) = φ([H, H]) = ˜ ′ . If ker(φ) ≤ H φ(⟨[u, v] | u, v ∈ H⟩) = ⟨[φ(u), φ(v)] | u, v ∈ H⟩ = [φ(H), φ(H)] = φ(H)′ = H ˜ ˜ , then φ can be restricted to an epimorphism φ|H : H → H , whence H = φ(H) ≃ H/ ker(φ) . Accord˜ H ˜ ′ from H onto the abelian ing to the previous section, the composite epimorphism (ωH˜ ◦ φ|H ) : H → H/ ′ ′ ˜ H ˜ factors through H/H by means of a uniquely determined epimorphism φ˜ : H/H ′ → H/ ˜ H ˜ ′ such group H/ ′ ′ ˜ H ˜ ≃ (H/H )/ ker(φ) that φ˜ ◦ ωH = ωH˜ ◦ φ|H . Consequently, we have H/ ˜ . Furthermore, the kernel of φ˜ is given explicitly by ker(φ) ˜ = ker(ωH˜ ◦ φ|H )/H ′ = (H ′ · ker(φ))/H ′ . Finally, if ker(φ) ≤ H ′ , then ˜ ′ = φ(H ′ ) ≃ H ′ / ker(φ) and φ˜ is an isomorphism, since ker(φ) H ˜ = H ′ /H ′ = 1 . Definition.
[9]
Due to the results in the present section, it makes sense to define a partial order on the set of abelian type invariants ˜ H ˜ ′ ⪯ H/H ′ , when H/ ˜ H ˜ ′ ≃ (H/H ′ )/ ker(φ) ˜ H ˜ ′ = H/H ′ , when H/ ˜ H ˜ ′ ≃ H/H ′ . by putting H/ ˜ , and H/
23.8.3
TKT singulets
˜ are groups, G ˜ = φ(G) is the image of G under an epimorphism φ : G → G ˜ , and H ˜ = φ(H) Suppose that G and G is the image of a subgroup H ≤ G of finite index n = (G : H) . Let TG,H be the Artin transfer from G to H/H ′ ˜ ˜ ˜′ and TG, ˜ H ˜ be the Artin transfer from G to H/H . If ker(φ) ≤ H , then the image (φ(g1 ), . . . , φ(gn )) of a left transversal (g1 , . . . , gn ) of H in G is a left transversal ˜ in G ˜ , and the inclusion φ(ker(TG,H )) ≤ ker(T ˜ ˜ ) holds. Moreover, if even ker(φ) ≤ H ′ , then the equation of H G,H φ(ker(TG,H )) = ker(TG, ˜ H ˜ ) holds. See Figure 3 for a visualization of this scenario. The truth of these statements can be justified in the following way. Let (g1 , . . . , gn ) be a left transversal of H in G . n n Then G = ∪˙ i=1 gi H is a disjoint union but φ(G) = ∪˙ i=1 φ(gi )φ(H) is not necessarily disjoint. For 1 ≤ j, k ≤ n , we have φ(gj )φ(H) = φ(gk )φ(H) ⇔ φ(H) = φ(gj )−1 φ(gk )φ(H) = φ(gj−1 gk )φ(H) ⇔ φ(gj−1 gk ) = φ(h) for some element h ∈ H ⇔ φ(h−1 gj−1 gk ) = 1 ⇔ h−1 gj−1 gk =: k ∈ ker(φ) . However, if the condition ker(φ) ≤ H is satisfied, then we are able to conclude that gj−1 gk = hk ∈ H , and thus j = k . ˜ H ˜ ′ be the epimorphism obtained in the manner indicated in the previous section. For the image Let φ˜ : H/H ′ → H/ ∏n ∏n xgi ·H ′ ) = i=1 φ(gπx (i) )−1 φ(x)φ(gi ))· of x ∈ G under the Artin transfer, we have φ(T ˜ G,H (x)) = φ( ˜ i=1 gπ−1 x (i) ˜ ′ , the right hand side equals T ˜ ˜ (φ(x)) , provided that (φ(g1 ), . . . , φ(gn )) φ(H ′ ) = . Since φ(H ′ ) = φ(H)′ = H G,H ˜ in G ˜ , which is correct, when ker(φ) ≤ H . This shows that the diagram in Figure 3 is is a left transversal of H commutative, that is φ˜ ◦ TG,H = TG, ˜ H ˜ ◦ φ . Consequently, we obtain the inclusion φ(ker(TG,H )) ≤ ker(TG, ˜ H ˜) , if ker(φ) ≤ H . Finally, if ker(φ) ≤ H ′ , then the previous section has shown that φ˜ is an isomorphism. Using the inverse isomorphism, we get TG,H = φ˜−1 ◦ TG, ˜ H ˜ ◦ φ , which proves the equation φ(ker(TG,H )) = ker(TG, ˜ H ˜) . Definition.
[9]
In view of the results in the present section, we are able to define a partial order of transfer kernels by setting ker(TG,H ) ⪯ ker(TG, ˜ H ˜ ) , when φ(ker(TG,H )) ≤ ker(TG, ˜ H ˜ ) , and ker(TG,H ) = ker(TG, ˜ H ˜ ) , when φ(ker(TG,H )) = ker(TG, ) . ˜ H ˜
23.8.4
TTT and TKT multiplets
˜ are groups, G ˜ = φ(G) is the image of G under an epimorphism φ : G → G ˜ , and both groups Suppose G and G ′ ′ ˜ ˜ have isomorphic finite abelianizations G/G ≃ G/G . Let (Hi )i∈I denote the family of all subgroups Hi ◁ G which contain the commutator subgroup G′ (and thus are necessarily normal), enumerated by means of the finite index set I , and let H˜i = φ(Hi ) be the image of Hi under φ , for each i ∈ I . Assume that, for each i ∈ I , Ti := TG,Hi ˜ denotes the Artin transfer from G to the abelianization Hi /Hi′ , and T˜i := TG, ˜ H ˜ i denotes the Artin transfer from G ˜ i /H ˜ ′ . Finally, let J ⊆ I be any non-empty subset of I . to the abelianization H i Then it is convenient to define κH (G) = (ker(Tj ))j∈J , called the (partial) transfer kernel type (TKT) of G with respect to (Hj )j∈J , and τH (G) = (Hj /Hj′ )j∈J , called the (partial) transfer target type (TTT) of G with respect to (Hj )j∈J . Due to the rules for singulets, established in the preceding two sections, these multiplets of TTTs and TKTs obey the following fundamental inheritance laws:
23.8. INHERITANCE FROM QUOTIENTS
Figure 3: Epimorphisms and Artin transfers.
˜ ⪯ τH (G) , in the sense that H ˜ j /H ˜ ′ ⪯ Hj /H ′ , for each 1. If ker(φ) ≤ ∩j∈J Hj , then τH˜ (G) j j
77
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CHAPTER 23. ARTIN TRANSFER (GROUP THEORY) ˜ , in the sense that ker(Tj ) ⪯ ker(T˜j ) , for each j ∈ J . j ∈ J , and κH (G) ⪯ κH˜ (G) ˜ = τH (G) , in the sense that H ˜ j /H ˜ ′ = Hj /H ′ , for each 2. If ker(φ) ≤ ∩j∈J Hj′ , then τH˜ (G) j j ˜ ˜ j ∈ J , and κH (G) = κH˜ (G) , in the sense that ker(Tj ) = ker(Tj ) , for each j ∈ J .
23.8.5
Inherited automorphisms
A further inheritance property does not immediately concern Artin transfers but will prove to be useful in applications to descendant trees. ˜ be groups such that G ˜ = φ(G) ≃ G/ ker(φ) is the image of G under an epimorphism φ : G → G ˜. Let G and G Suppose that σ ∈ Aut(G) is an automorphism of G . ˜ → G ˜ such that φ ◦ σ = σ If σ(ker(φ)) ≤ ker(φ) , then there exists a unique epimorphism σ ˜ : G ˜ ◦ φ . If ˜ is also an automorphism. σ(ker(φ)) = ker(φ) , then σ ˜ ∈ Aut(G) ˜ = φ(G) ≃ G/ ker(φ) , which permits The justification for these facts is based on the isomorphic representation G to identify σ ˜ (g ker(φ))=˜ ˆ σ (φ(g)) = φ(σ(g))=σ(g) ˆ ker(φ) for all g ∈ G and proves the uniqueness of σ ˜ . If σ(ker(φ)) ≤ ker(φ) , then the consistency follows from g ker(φ) = h ker(φ) ⇒ h−1 g ∈ ker(φ) ⇒ σ(h−1 g) ∈ ker(φ) ⇒ σ(g) ker(φ) = σ(h) ker(φ) . And if σ(ker(φ)) = ker(φ) , then injectivity of σ ˜ is a consequence of σ ˜ (g ker(φ))=σ(g) ˆ ker(φ) = ker(φ) ⇒ σ(g) ∈ ker(φ) ⇒ g = σ −1 (σ(g)) ∈ ker(φ) , since σ −1 (ker(φ)) ≤ ker(φ) . Now, let us denote the canonical projection from G to its abelianization G/G′ by ω : G → G/G′ . There exists a ¯ ∈ Aut(G/G′ ) such that ω ◦ σ = σ ¯ ◦ ω , that is, σ ¯ (gG′ ) = σ ¯ (ω(g)) = ω(σ(g)) = unique induced automorphism σ ′ ′ σ(g)G , for all g ∈ G . The reason for the injectivity of σ ¯ is that σ(g)G = σ ¯ (gG′ ) = G′ ⇒ σ(g) ∈ G′ ⇒ g = σ −1 (σ(g)) ∈ G′ , since G′ is a characteristic subgroup of G . Definition. G is called a σ-group, if there exists an automorphism σ ∈ Aut(G) such that the induced automorphism acts like the inversion on G/G′ , that is, σ(g)G′ = σ ¯ (gG′ ) = g −1 G′ , resp. σ(g) ≡ g −1 (mod G′ ) , for all g ∈ G . ˜ is also a The supplementary inheritance property asserts that, if G is a σ -group and σ(ker(φ)) = ker(φ) , then G σ -group, the required automorphism being σ ˜. This can be seen by applying the epimorphism φ to the equation σ(g)G′ = σ ¯ (gG′ ) = g −1 G′ , for g ∈ G , which ˜ ′ , for all x = φ(g) ∈ ˜ ′ = x−1 G ˜′ = σ ˜ ′ = φ(σ(g))φ(G′ ) = φ(g −1 )φ(G′ ) = φ(g)−1 G yields σ ˜ (x)G ˜ (φ(g))G ˜. φ(G) = G
23.9 Stabilization criteria In this section, the results concerning the inheritance of TTTs and TKTs from quotients in the previous section are applied to the simplest case, which is characterized by the following Assumption. The parent π(G) of a group G is the quotient π(G) = G/N of G by the last non-trivial term N = γc (G) ◁ G of the lower central series of G , where c denotes the nilpotency class of G . The corresponding epimorphism π from G onto π(G) = G/γc (G) is the canonical projection, whose kernel is given by ker(π) = γc (G) . Under this assumption, the kernels and targets of Artin transfers turn out to be compatible with parent-descendant relations between finite p-groups. Compatibility criterion. Let p be a prime number. Suppose that G is a non-abelian finite p-group of nilpotency class c = cl(G) ≥ 2 . Then the TTT and the TKT of G and of its parent π(G) are comparable in the sense that τ (π(G)) ⪯ τ (G) and κ(G) ⪯ κ(π(G)) . The simple reason for this fact is that, for any subgroup G′ ≤ H ≤ G , we have ker(π) = γc (G) ≤ γ2 (G) = G′ ≤ H , since c ≥ 2 . For the remaining part of this section, the investigated groups are supposed to be finite metabelian p-groups G with elementary abelianization G/G′ of rank 2 , that is of type (p, p) .
23.9. STABILIZATION CRITERIA
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Partial stabilization for maximal class. A metabelian p-group G of coclass cc(G) = 1 and of nilpotency class c = cl(G) ≥ 3 shares the last p components of the TTT τ (G) and of the TKT κ(G) with its parent π(G) . More explicitly, for odd primes p ≥ 3 , we have τ (G)i = (p, p) and κ(G)i = 0 for 2 ≤ i ≤ p + 1 . [10] This criterion is due to the fact that c ≥ 3 implies ker(π) = γc (G) ≤ γ3 (G) = Hi′ , subgroups H2 , . . . , Hp+1 of G .
[11]
for the last p maximal
The condition c ≥ 3 is indeed necessary for the partial stabilization criterion. For odd primes p ≥ 3 , the extra special p -group G = G30 (0, 1) of order p3 and exponent p2 has nilpotency class c = 2 only, and the last p components of its TKT κ = (1p+1 ) are strictly smaller than the corresponding components of the TKT κ = (0p+1 ) of its parent π(G) which is the elementary abelian p -group of type (p, p) . [10] For p = 2 , both extra special 2 -groups of coclass 1 and class c = 2 , the ordinary quaternion group G = G30 (0, 1) with TKT κ = (123) and the dihedral group G = G30 (0, 0) with TKT κ = (023) , have strictly smaller last two components of their TKTs than their common parent π(G) = C2 × C2 with TKT κ = (000) . Total stabilization for maximal class and positive defect. A metabelian p-group G of coclass cc(G) = 1 and of nilpotency class c = m − 1 = cl(G) ≥ 4 , that is, with index of nilpotency m ≥ 5 , shares all p + 1 components of the TTT τ (G) and of the TKT κ(G) with its parent π(G) , provided it has positive defect of commutativity k = k(G) ≥ 1 . [5] Note that k ≥ 1 implies p ≥ 3 , and we have κ(G)i = 0 for all 1 ≤ i ≤ p + 1 . [10] This statement can be seen by observing that the conditions m ≥ 5 and k ≥ 1 imply ker(π) = γm−1 (G) ≤ γm−k (G) ≤ Hi′ , [11] for all the p + 1 maximal subgroups H1 , . . . , Hp+1 of G . The condition k ≥ 1 is indeed necessary for total stabilization. To see this it suffices to consider the first component of the TKT only. For each nilpotency class c ≥ 4 , there exist (at least) two groups G = Gc+1 0 (0, 1) with TKT p κ = (10p ) and G = Gc+1 (1, 0) with TKT κ = (20 ) , both with defect k = 0 , where the first component of their 0 TKT is strictly smaller than the first component of the TKT κ = (0p+1 ) of their common parent π(G) = Gc0 (0, 0) . Partial stabilization for non-maximal class. Let p = 3 be fixed. A metabelian 3-group G with abelianization G/G′ ≃ (3, 3) , coclass cc(G) ≥ 2 and nilpotency class c = cl(G) ≥ 4 shares the last two (among the four) components of the TTT τ (G) and of the TKT κ(G) with its parent π(G) . This criterion is justified by the following consideration. If c ≥ 4 , then ker(π) = γc (G) ≤ γ4 (G) ≤ Hi′ [11] for the last two maximal subgroups H3 , H4 of G . The condition c ≥ 4 is indeed unavoidable for partial stabilization, since there exist several 3 -groups of class c = 3 , for instance those with SmallGroups identifiers G ∈ {⟨243, 3⟩, ⟨243, 6⟩, ⟨243, 8⟩} , such that the last two components of their TKTs κ ∈ {(0043), (0122), (2034)} are strictly smaller than the last two components of the TKT κ = (0000) of their common parent π(G) = G30 (0, 0) . Total stabilization for non-maximal class and cyclic centre. Again, let p = 3 be fixed. A metabelian 3-group G with abelianization G/G′ ≃ (3, 3) , coclass cc(G) ≥ 2 , nilpotency class c = cl(G) ≥ 4 and cyclic centre ζ1 (G) shares all four components of the TTT τ (G) and of the TKT κ(G) with its parent π(G) . The reason is that, due to the cyclic centre, we have ker(π) = γc (G) = ζ1 (G) ≤ Hi′ subgroups H1 , . . . , H4 of G .
[11]
for all four maximal
The condition of a cyclic centre is indeed necessary for total stabilization, since for a group with bicyclic centre there occur two possibilities. Either γc (G) = ζ1 (G) is also bicyclic, whence γc (G) is never contained in H2′ , or γc (G) < ζ1 (G) is cyclic but is never contained in H1′ . Summarizing, we can say that the last four criteria underpin the fact that Artin transfers provide a marvellous tool for classifying finite p-groups. In the following sections, it will be shown how these ideas can be applied for endowing descendant trees with additional structure, and for searching particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in pure group theory and in algebraic number theory.
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Figure 4: Endowing a descendant tree with information on Artin transfers.
23.10 Structured descendant trees (SDTs) This section uses the terminology of descendant trees in the theory of finite p-groups. In Figure 4, a descendant tree with modest complexity is selected exemplarily to demonstrate how Artin transfers provide additional structure for each vertex of the tree. More precisely, the underlying prime is p = 3 , and the chosen descendant tree is actually a coclass tree having a unique infinite mainline, branches of depth 3 , and strict periodicity of length 2 setting in with branch B(7) . The initial pre-period consists of branches B(5) and B(6) with exceptional structure. Branches B(7) and B(8) form the primitive period such that B(j) ≃ B(7) , for odd j ≥ 9 , and B(j) ≃ B(8) , for even j ≥ 10 . The root of the tree is the metabelian 3 -group with identifier R = ⟨243, 6⟩ , that is, a group of order |R| = 35 = 243 and with counting number 6 . It should be emphasized that this root is not coclass settled, whence its entire descendant tree T (R) is of considerably higher complexity than the coclass- 2 subtree T 2 (R) , whose first six branches are drawn in the diagram of Figure 4. The additional structure can be viewed as a sort of coordinate system in which the tree is embedded. The horizontal abscissa is labelled with the transfer kernel type (TKT) κ , and the vertical ordinate is labelled with a single component τ (1) of the transfer target type (TTT). The vertices of the tree are drawn in such a manner that members of periodic infinite sequences form a vertical column sharing a common TKT. On the other hand, metabelian groups of a fixed order, represented by vertices of depth at most 1 , form a horizontal row sharing a common first component of the TTT. (To discourage any incorrect interpretations, we explicitly point out that the first component of the TTT of non-metabelian groups or metabelian groups, represented by vertices of depth 2 , is usually smaller than expected, due to stabilization phenomena!) The TTT of all groups in this tree represented
23.11. PATTERN RECOGNITION
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by a big full disk, which indicates a bicyclic centre of type (3, 3) , is given by τ = [A(3, c), (3, 3, 3), (9, 3), (9, 3)] with varying first component τ (1) = A(3, c) , the nearly homocyclic abelian 3 -group of order 3c , and fixed further components τ (2) = (3, 3, 3)=(1 ˆ 3 ) and τ (3) = τ (4) = (9, 3)=(21) ˆ , where the abelian type invariants are either written as orders of cyclic components or as their 3 -logarithms with exponents indicating iteration. (The latter notation is employed in Figure 4.) Since the coclass of all groups in this tree is 2 , the connection between the order 3n and the nilpotency class is given by c = n − 2 .
23.11 Pattern recognition For searching a particular group in a descendant tree by looking for patterns defined by the kernels and targets of Artin transfers, it is frequently adequate to reduce the number of vertices in the branches of a dense tree with high complexity by sifting groups with desired special properties, for example • filtering the σ -groups, • eliminating a set of certain transfer kernel types, • cancelling all non-metabelian groups (indicated by small contour squares in Fig. 4), • removing metabelian groups with cyclic centre (denoted by small full disks in Fig. 4), • cutting off vertices whose distance from the mainline (depth) exceeds some lower bound, • combining several different sifting criteria. The result of such a sieving procedure is called a pruned descendant tree with respect to the desired set of properties. However, in any case, it should be avoided that the main line of a coclass tree is eliminated, since the result would be a disconnected infinite set of finite graphs instead of a tree. For example, it is neither recommended to eliminate all σ -groups in Figure 4 nor to eliminate all groups with TKT κ = (0122) . In Figure 4, the big double contour rectangle surrounds the pruned coclass tree T∗2 (R) , where the numerous vertices with TKT κ = (2122) are completely eliminated. This would, for instance, be useful for searching a σ -group with TKT κ = (1122) and first component τ (1) = (43) of the TTT. In this case, the search result would even be a unique group. We expand this idea further in the following detailed discussion of an important example.
23.11.1
Historical example
The oldest example of searching for a finite p-group by the strategy of pattern recognition via Artin transfers goes back to 1934, when A. Scholz and O. Taussky [12] tried to determine the Galois group G = G∞ 3 (K) = ∞ (K) , of the tower, that is the maximal unramified pro3 extension F Gal(F∞ 3 3 (K)|K) of the Hilbert 3 -class field √ complex quadratic number field K = Q( −9748) . They actually succeeded in finding the maximal metabelian quotient Q = G/G′′ = G23 (K) = Gal(F23 (K)|K) of G , that is the Galois group of the second Hilbert 3 -class field F23 (K) of K . However, it needed 78 years until M. R. Bush and D. C. Mayer, in 2012, provided the first rigorous proof [9] that the (potentially infinite) 3 -tower group G = G∞ 3 (K) coincides with the finite 3 -group G33 (K) = Gal(F33 (K)|K) of derived length dl(G) = 3 , and thus the 3 -tower of K has exactly three stages, stopping at the third Hilbert 3 -class field F33 (K) of K . The search is performed with the aid of the p-group generation algorithm by M. F. Newman [13] and E. A. O'Brien. For the initialization of the algorithm, two basic invariants must be determined. Firstly, the generator rank d of the p-groups to be constructed. Here, we have p = 3 and d = r3 (K) = d(Cl3 (K)) is given by the 3 -class rank of the quadratic field K . Secondly, the abelian type invariants of the 3 -class group Cl3 (K) ≃ (12 ) of K . These two invariants indicate the root of the descendant tree which will be constructed successively. Although the p-group generation algorithm is designed to use the parent-descendant definition by means of the lower exponent-p central series, it can be fitted to the definition with the aid of the usual lower central series. In the case of an elementary abelian p-group as root, the difference is not very big. So we have to start with the elementary abelian 3 -group of rank two, which has the SmallGroups identifier ⟨9, 2⟩ , and to construct the descendant tree T (⟨9, 2⟩) . We do that by iterating the p-group generation algorithm, taking suitable capable descendants of the previous root as the next root, always executing an increment of the nilpotency class by a unit. [14]
As explained at the beginning of the section Pattern recognition, we must prune the descendant tree with respect to the invariants TKT and TTT of the 3 -tower group G , which are determined by the arithmetic of the field K as
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κ ∈ {(2334), (2434)} (exactly two fixed points and no transposition) and τ = [(21)(32)(21)(21)] . Further, any quotient of G must be a σ -group, enforced by number theoretic requirements for the quadratic field K . The root ⟨9, 2⟩ has only a single capable descendant ⟨27, 3⟩ of type (12 ) . In terms of the nilpotency class, ⟨9, 2⟩ is the class- 1 quotient G/γ2 (G) of G and ⟨27, 3⟩ is the class- 2 quotient G/γ3 (G) of G . Since the latter has nuclear rank ˙ 2 (⟨27, 3⟩) , where the former component T 1 (⟨27, 3⟩) two, there occurs a bifurcation T (⟨27, 3⟩) = T 1 (⟨27, 3⟩)∪T can be eliminated by the stabilization criterion κ = (∗000) for the TKT of all 3 -groups of maximal class. Due to the inheritance property of TKTs, only the single capable descendant ⟨243, 8⟩ qualifies as the class- 3 quotient G/γ4 (G) of G . There is only a single capable σ -group ⟨729, 54⟩ among the descendants of ⟨243, 8⟩ . It is the class4 quotient G/γ5 (G) of G and has nuclear rank two. ˙ 3 (⟨729, 54⟩) in two subtrees belonging to This causes the essential bifurcation T (⟨729, 54⟩) = T 2 (⟨729, 54⟩)∪T different coclass graphs G(3, 2) and G(3, 3) . The former contains the metabelian quotient Q = G/G′′ of G with two possibilities Q ∈ {⟨2187, 302⟩, ⟨2187, 306⟩} , which are not balanced with relation rank r = 3 > 2 = d bigger than the generator rank. The latter consists entirely of non-metabelian groups and yields the desired 3 -tower group G as one among the two Schur σ -groups ⟨729, 54⟩ − #2; 2 and ⟨729, 54⟩ − #2; 6 with r = 2 = d . Finally the termination criterion is reached at the capable vertices ⟨2187, 303⟩ − #1; 1 ∈ G(3, 2) and ⟨729, 54⟩ − #2; 3 − #1; 1 ∈ G(3, 3) , since the TTT τ = [(21)(32 )(21)(21)] > [(21)(32)(21)(21)] is too big and will even increase further, never returning back to [(21)(32)(21)(21)] . The complete search process is visualized in Table 1, where, for√each of the possible successive p-quotients Pc = G/γc+1 (G) of the 3 -tower group G = G∞ 3 (K) of K = Q( −9748) , the nilpotency class is denoted by c = cl(Pc ) , the nuclear rank by ν = ν(Pc ) , and the p-multiplicator rank by µ = µ(Pc ) .
23.12 Commutator calculus This section shows exemplarily how commutator calculus can be used for determining the kernels and targets of Artin transfers explicitly. As a concrete example we take the metabelian 3 -groups with bicyclic centre, which are represented by big full disks as vertices, of the coclass tree diagram in Figure 4. They form ten periodic infinite sequences, four, resp. six, for even, resp. odd, nilpotency class c , and can be characterized with the aid of a parametrized polycyclic power-commutator presentation: Gc,n (z, w) =⟨x, y, s2 , t3 , s3 , . . . , sc | 1
3 2 3 2 3 z 3 2 x3 = sw c , y = s3 s4 sc , sj = sj+2 sj+3 for 2 ≤ j ≤ c − 3, sc−2 = sc , t3 = 1,
s2 = [y, x], t3 = [s2 , y], sj = [sj−1 , x] for 3 ≤ j ≤ c⟩, where c ≥ 5 is the nilpotency class, 3n with n = c + 2 is the order, and 0 ≤ w ≤ 1 , −1 ≤ z ≤ 1 are parameters. The transfer target type (TTT) of the group G = Gc,n (z, w) depends only on the nilpotency class c , is independent of the parameters w, z , and is given uniformly by τ = [A(3, c), (3, 3, 3), (9, 3), (9, 3)] . This phenomenon is called a polarization, more precisely a uni-polarization, [5] at the first component. The transfer kernel type (TKT) of the group G = Gc,n (z, w) is independent of the nilpotency class c , but depends on the parameters w, z , and is given by c.18, κ = (0122) , for w = z = 0 (a mainline group), H.4, κ = (2122) , for w = 0, z = ±1 (two capable groups), E.6, κ = (1122) , for w = 1, z = 0 (a terminal group), and E.14, κ ∈ {(4122), (3122)} , for w = 1, z = ±1 (two terminal groups). For even nilpotency class, the two groups of types H.4 and E.14, which differ in the sign of the parameter z only, are isomorphic. These statements can be deduced by means of the following considerations. As a preparation, it is useful to compile a list of some commutator relations, starting with those given in the presentation, [a, x] = 1 for a ∈ {sc , t3 } and [a, y] = 1 for a ∈ {s3 , . . . , sc , t3 } , which shows that the bicyclic centre is given by ζ1 (G) = ⟨sc , t3 ⟩ . By means of the right product rule [a, xy] = [a, y] · [a, x] · [[a, x], y] and the right power rule [a, y 2 ] = [a, y]1+y , we obtain [s2 , xy] = s3 t3 , [s2 , xy 2 ] = s3 t23 , and [sj , xy] = [sj , xy 2 ] = [sj , x] = sj+1 , for j ≥ 3 . The maximal subgroups of G are taken in a similar way as in the section on the computational implementation, namely H1 = ⟨y, G′ ⟩ , H2 = ⟨x, G′ ⟩ , H3 = ⟨xy, G′ ⟩ , and H4 = ⟨xy 2 , G′ ⟩ . Their derived subgroups are crucial for the behavior of the Artin transfers. By making use of the general formula Hi′ = (G′ )hi −1 , where Hi = ⟨hi , G′ ⟩ , and where we know that G′ = ⟨s2 , t3 , s3 , . . . , sc ⟩ in the present situation, xy−1 ′ it follows that H1′ = ⟨sy−1 ⟩ = ⟨t3 ⟩ , H2′ = ⟨sx−1 , . . . , sx−1 , . . . , sxy−1 2 2 c−1 ⟩ = ⟨s3 , . . . , sc ⟩ , H3 = ⟨s2 c−1 ⟩ =
23.13. SYSTEMATIC LIBRARY OF SDTS
83
−1 −1 2 ⟨s3 t3 , s4 , . . . , sc ⟩ , and H4′ = ⟨sxy , . . . , sxy 2 c−1 ⟩ = ⟨s3 t3 , s4 , . . . , sc ⟩ . Note that H1 is not far from being ′ abelian, since H1 = ⟨t3 ⟩ is contained in the centre ζ1 (G) = ⟨sc , t3 ⟩ . 2
2
As the first main result, we are now in the position to determine the abelian type invariants of the derived quotients: H1 /H1′ = ⟨y, s2 , . . . , sc ⟩H1′ /H1′ ≃ A(3, c) , the unique quotient which grows with increasing nilpotency class c , since ord(y) = ord(s2 ) = 3m for even c = 2m and ord(y) = 3m+1 , ord(s2 ) = 3m for odd c = 2m + 1 , H2 /H2′ = ⟨x, s2 , t3 ⟩H2′ /H2′ ≃ (3, 3, 3) , H3 /H3′ = ⟨xy, s2 , t3 ⟩H3′ /H3′ ≃ (9, 3) , H4 /H4′ = ⟨xy 2 , s2 , t3 ⟩H4′ /H4′ ≃ (9, 3) , since generally ord(s2 ) = ord(t3 ) = 3 , but ord(x) = 3 for H2 , whereas ord(xy) = ord(xy 2 ) = 9 for H3 and H4 . Now we come to the kernels of the Artin transfer homomorphisms Ti : G → Hi /Hi′ . It suffices to investigate the induced transfers T˜i : G/G′ → Hi /Hi′ and to begin by finding expressions for the images T˜i (gG′ ) of elements gG′ ∈ G/G′ , which can be expressed in the form g ≡ xj y ℓ (mod G′ ) with exponents −1 ≤ j, ℓ ≤ 1 . First, ′ we exploit outer transfers as much as possible: x ∈ / H1 ⇒ T˜1 (xG′ ) = x3 H1′ = sw / H2 ⇒ T˜2 (yG′ ) = c H1 , y ∈ 3 ′ 2 z ′ ′ ′ 3 ′ w ′ ′ ˜ y H2 = s3 s4 sc H2 = 1 · H2 , x, y ∈ / H3 , H4 ⇒ Ti (xG ) = x Hi = sc Hi = 1 · Hi and T˜i (yG′ ) = y 3 Hi′ = 2 z ′ 2 ′ s3 s4 sc Hi = s3 Hi , for 3 ≤ i ≤ 4 . Next, we treat the unavoidable inner transfers, which are more intricate. For this purpose, we use the polynomial identity X 2 +X +1 = (X −1)2 +3(X −1)+3 to obtain: y ∈ H1 ⇒ T˜1 (yG′ ) = 2 2 y 1+x+x H1′ = y 3+3(x−1)+(x−1) H1′ = y 3 · [y, x]3 · [[y, x], x]H1′ = s23 s4 szc s32 s3 H1′ = s32 s33 s4 szc H1′ = szc H1′ and 2 2 −3 −1 ′ −1 ′ x ∈ H2 ⇒ T˜2 (xG′ ) = x1+y+y H2′ = x3+3(y−1)+(y−1) H2′ = x3 ·[x, y]3 ·[[x, y], y]H2′ = sw c s2 t3 H2 = t3 H2 ′ ′ j ′ ℓ ′ wj+zℓ . Finally, we combine the results: generally T˜i (gG ) = T˜i (xG ) T˜i (yG ) , and in particular, T˜1 (gG ) = sc H1′ −j ′ ′ ′ 2ℓ ′ , T˜2 (gG ) = t3 H2 , T˜i (gG ) = s3 Hi , for 3 ≤ i ≤ 4 . To determine the kernels, it remains to solve the following equations: swj+zℓ H1′ = H1′ ⇒ j, ℓ arbitrary for w = z = 0 , ℓ = 0 with arbitrary j for w = 0, z = ±1 , j = 0 with c ′ ′ arbitrary ℓ for w = 1, z = 0 , and j = ∓ℓ for w = 1, z = ±1 , furthermore, t−j 3 H2 = H2 ⇒ j = 0 with arbitrary ′ 2ℓ ′ ℓ , s3 Hi = Hi ⇒ ℓ = 0 with arbitrary j , for 3 ≤ i ≤ 4 . The following equivalences, for any 1 ≤ i ≤ 4 , finish the justification of the statements: j = 0 with arbitrary ℓ ⇔ ker(Ti ) = ⟨y, G′ ⟩ = H1 ⇔ κ(i) = 1 , ℓ = 0 with arbitrary j ⇔ ker(Ti ) = ⟨x, G′ ⟩ = H2 ⇔ κ(i) = 2 , j = ℓ ⇔ ker(Ti ) = ⟨xy, G′ ⟩ = H3 ⇔ κ(i) = 3 , j = −ℓ ⇔ ker(Ti ) = ⟨xy −1 , G′ ⟩ = H4 ⇔ κ(i) = 4 , and j, ℓ both arbitrary ⇔ ker(Ti ) = ⟨x, y, G′ ⟩ = G ⇔ κ(i) = 0 . Consequently, the last three components of the TKT are independent of the parameters w, z , which means that both, the TTT and the TKT, reveal a uni-polarization at the first component.
23.13 Systematic library of SDTs The aim of this section is to present a collection of structured coclass trees (SCTs) of finite p-groups with parametrized presentations and a succinct summary of invariants. The underlying prime p is restricted to small values p ∈ {2, 3, 5} . The trees are arranged according to increasing coclass r ≥ 1 and different abelianizations within each coclass. To keep the descendant numbers manageable, the trees are pruned by eliminating vertices of depth bigger than one. Further, we omit trees where stabilization criteria enforce a common TKT of all vertices, since we do not consider such trees as structured any more. The invariants listed include • pre-period and period length, • depth and width of branches, • uni-polarization, TTT and TKT, • σ -groups. We refrain from giving justifications for invariants, since the way how invariants are derived from presentations was demonstrated exemplarily in the section on commutator calculus
23.13.1
Coclass 1
For each prime p ∈ {2, 3, 5} , the unique tree of p-groups of maximal class is endowed with information on TTTs and TKTs, that is, T 1 (⟨4, 2⟩) for p = 2 , T 1 (⟨9, 2⟩) for p = 3 , and T 1 (⟨25, 2⟩) for p = 5 . In the last case, the tree is restricted to metabelian 5 -groups. The 2 -groups of coclass 1 in Figure 5 can be defined by the following parametrized polycyclic pc-presentation, quite different from Blackburn’s presentation. [4]
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Figure 5: Structured descendant tree of 2-groups with coclass 1.
Gc,n (z, w) =⟨x, y, s2 , . . . , sc | 2
2 z 2 2 x2 = sw c , y = sc , sj = sj+1 sj+2 for 2 ≤ j ≤ c − 2, sc−1 = sc ,
s2 = [y, x], sj = [sj−1 , x] = [sj−1 , y] for 3 ≤ j ≤ c⟩, where the nilpotency class is c ≥ 3 , the order is 2n with n = c + 1 , and w, z are parameters. The branches are strictly periodic with pre-period 1 and period length 1 , and have depth 1 and width 3 . Polarization occurs for the third component and the TTT is τ = [(12 ), (12 ), A(2, c)] , only dependent on c and with cyclic A(2, c) . The TKT depends on the parameters and is κ = (210) for the dihedral mainline vertices with w = z = 0 , κ = (213) for the terminal generalized quaternion groups with w = z = 1 , and κ = (211) for the terminal semi dihedral groups with w = 1, z = 0 . There are two exceptions, the abelian root with τ = [(1), (1), (1)] and κ = (000) , and the usual quaternion group with τ = [(2), (2), (2)] and κ = (123) . The 3 -groups of coclass 1 in Figure 6 can be defined by the following parametrized polycyclic pc-presentation, slightly different from Blackburn’s presentation. [4]
23.13. SYSTEMATIC LIBRARY OF SDTS
85
Figure 6: Structured descendant tree of 3-groups with coclass 1.
Gc,n a (z, w) =⟨x, y, s2 , t3 , s3 , . . . , sc | 3
3 2 z a 3 2 3 2 x3 = sw c , y = s3 s4 sc , t3 = sc , sj = sj+2 sj+3 for 2 ≤ j ≤ c − 3, sc−2 = sc ,
s2 = [y, x], t3 = [s2 , y], sj = [sj−1 , x] for 3 ≤ j ≤ c⟩, where the nilpotency class is c ≥ 5 , the order is 3n with n = c + 1 , and a, w, z are parameters. The branches are strictly periodic with pre-period 2 and period length 2 , and have depth 1 and width 7 . Polarization occurs for the first component and the TTT is τ = [A(3, c − a), (12 ), (12 ), (12 )] , only dependent on c and a . The TKT depends on the parameters and is κ = (0000) for the mainline vertices with a = w = z = 0 , κ = (1000) for the terminal vertices with a = 0, w = 1, z = 0 , κ = (2000) for the terminal vertices with a = w = 0, z = ±1 , and κ = (0000) for the terminal vertices with a = 1, w ∈ {−1, 0, 1}, z = 0 . There exist three exceptions, the abelian root with τ = [(1), (1), (1), (1)] , the extra special group of exponent 9 with τ = [(12 ), (2), (2), (2)] and κ = (1111) , and the Sylow 3 -subgroup of the alternating group A9 with τ = [(13 ), (12 ), (12 ), (12 )] . Mainline vertices and vertices on odd branches are σ -groups. The metabelian 5 -groups of coclass 1 in Figure 7 can be defined by the following parametrized polycyclic pcpresentation, slightly different from Miech’s presentation. [15] Gc,n a (z, w) =⟨x, y, s2 , t3 , s3 , . . . , sc | 4
5 z a x5 = sw c , y = sc , t3 = sc , s2 = [y, x], t3 = [s2 , y], sj = [sj−1 , x] for 3 ≤ j ≤ c⟩,
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Figure 7: Structured descendant tree of metabelian 5-groups with coclass 1.
where the nilpotency class is c ≥ 3 , the order is 5n with n = c + 1 , and a, w, z are parameters. The (metabelian!) branches are strictly periodic with pre-period 3 and period length 4 , and have depth 3 and width 67 . (The branches of the complete tree, including non-metabelian groups, are only virtually periodic and have bounded width but unbounded depth!) Polarization occurs for the first component and the TTT is τ = [A(5, c−k), (12 )5 ] , only dependent on c and the defect of commutativity k . The TKT depends on the parameters and is κ = (06 ) for the mainline vertices with a = w = z = 0 , κ = (105 ) for the terminal vertices with a = 0, w = 1, z = 0 , κ = (205 ) for the terminal vertices with a = w = 0, z ̸= 0 , and κ = (06 ) for the vertices with a ̸= 0 . There exist three exceptions, the abelian root with τ = [(1)6 ] , the extra special group of exponent 25 with τ = [(12 ), (2)5 ] and κ = (16 ) , and the group ⟨15625, 631⟩ with τ = [(15 ), (12 )5 ] . Mainline vertices and vertices on odd branches are σ -groups.
23.13.2
Coclass 2
Abelianization of type (p,p) Three coclass trees, T 2 (⟨243, 6⟩) , T 2 (⟨243, 8⟩) and T 2 (⟨729, 40⟩) for p = 3 , are endowed with information concerning TTTs and TKTs. On the tree T 2 (⟨243, 6⟩) , the 3 -groups of coclass 2 with bicyclic centre in Figure 8 can be defined by the following
23.13. SYSTEMATIC LIBRARY OF SDTS
87
Figure 8: First structured descendant tree of 3-groups with coclass 2 and abelianization (3,3).
parametrized polycyclic pc-presentation.
[5]
Gc,n (z, w) =⟨x, y, s2 , t3 , s3 , . . . , sc | 5
3 2 z 3 2 3 2 3 x3 = sw c , y = s3 s4 sc , sj = sj+2 sj+3 for 2 ≤ j ≤ c − 3, sc−2 = sc , t3 = 1,
s2 = [y, x], t3 = [s2 , y], sj = [sj−1 , x] for 3 ≤ j ≤ c⟩, where the nilpotency class is c ≥ 5 , the order is 3n with n = c + 2 , and w, z are parameters. The branches are strictly periodic with pre-period 2 and period length 2 , and have depth 3 and width 18 . Polarization occurs for the first component and the TTT is τ = [A(3, c), (13 ), (21), (21)] , only dependent on c . The TKT depends on the parameters and is κ = (0122) for the mainline vertices with w = z = 0 , κ = (2122) for the capable vertices with w = 0, z = ±1 , κ = (1122) for the terminal vertices with w = 1, z = 0 , and κ = (3122) for the terminal vertices with w = 1, z = ±1 . Mainline vertices and vertices on even branches are σ -groups. On the tree T 2 (⟨243, 8⟩) , the 3 -groups of coclass 2 with bicyclic centre in Figure 9 can be defined by the following parametrized polycyclic pc-presentation. [5] Gc,n (z, w) =⟨x, y, t2 , s3 , t3 , . . . , tc | 6
3 2 z 3 2 3 2 3 y 3 = s3 tw c , x = t3 t4 t5 tc , tj = tj+2 tj+3 for 2 ≤ j ≤ c − 3, tc−2 = tc , s3 = 1,
t2 = [y, x], s3 = [t2 , x], tj = [tj−1 , y] for 3 ≤ j ≤ c⟩, where the nilpotency class is c ≥ 6 , the order is 3n with n = c + 2 , and w, z are parameters. The branches are strictly periodic with pre-period 2 and period length 2 , and have depth 3 and width 16 . Polarization occurs for the second component and the TTT is τ = [(21), A(3, c), (21), (21)] , only dependent on c . The TKT depends on the parameters and is κ = (2034) for the mainline vertices with w = z = 0 , κ = (2134) for the capable vertices with
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CHAPTER 23. ARTIN TRANSFER (GROUP THEORY)
Figure 9: Second structured descendant tree of 3-groups with coclass 2 and abelianization (3,3).
w = 0, z = ±1 , κ = (2234) for the terminal vertices with w = 1, z = 0 , and κ = (2334) for the terminal vertices with w = 1, z = ±1 . Mainline vertices and vertices on even branches are σ -groups.
Abelianization of type (p2 ,p) T 2 (⟨16, 3⟩) and T 2 (⟨16, 4⟩) for p = 2 , T 2 (⟨243, 15⟩) and T 2 (⟨243, 17⟩) for p = 3 .
Abelianization of type (p,p,p) T 2 (⟨16, 11⟩) for p = 2 , and T 2 (⟨81, 12⟩) for p = 3 .
23.13.3
Coclass 3
Abelianization of type (p2 ,p) T 3 (⟨729, 13⟩) , T 3 (⟨729, 18⟩) and T 3 (⟨729, 21⟩) for p = 3 .
23.13. SYSTEMATIC LIBRARY OF SDTS Abelianization of type (p,p,p) T 3 (⟨32, 35⟩) and T 3 (⟨64, 181⟩) for p = 2 , T 3 (⟨243, 38⟩) and T 3 (⟨243, 41⟩) for p = 3 .
Figure 10: Minimal discriminants for the first ASCT of 3-groups with coclass 2 and abelianization (3,3).
89
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23.14 Arithmetical applications In algebraic number theory and class field theory, structured descendant trees (SDTs) of finite p-groups provide an excellent tool for • visualizing the location of various non-abelian p-groups G(K) associated with algebraic number fields K , • displaying additional information about the groups G(K) in labels attached to corresponding vertices, and • emphasizing the periodicity of occurrences of the groups G(K) on branches of coclass trees. For instance, let p be a prime number, and assume that Fp2 (K) denotes the second Hilbert p-class field of an algebraic number field K , that is the maximal metabelian unramified extension of K of degree a power of p . Then the second p-class group G2p (K) = Gal(Fp2 (K)|K) of K is usually a non-abelian p-group of derived length 2 and frequently permits to draw conclusions about the entire p-class field tower of K , that is the Galois group ∞ ∞ G∞ p (K) = Gal(Fp (K)|K) of the maximal unramified pro-p extension Fp (K) of K . Given a sequence of algebraic number fields K with fixed signature (r1 , r2 ) , ordered by the absolute values |d| of their discriminants d = d(K) , a suitable structured coclass tree (SCT) T , or also the finite sporadic part G0 (p, r) of a coclass graph G(p, r) , whose vertices are entirely or partially realized by second p-class groups G2p (K) of the fields K is endowed with additional arithmetical structure when each realized vertex V ∈ T , resp. V ∈ G0 (p, r) , is mapped to data concerning the fields K such that V = G2p (K) .
23.14.1
Example
√ To be specific, let p = 3 and consider complex quadratic fields K(d) = Q( d) with fixed signature (0, 1) having 3 -class groups with type invariants (3, 3) . See OEIS A242863 . Their second 3 -class groups G23 (K) have been determined by D. C. Mayer [11] for the range −106 < d < 0 , and, most recently, by N. Boston, M. R. Bush and F. Hajir [16] for the extended range −108 < d < 0 . Let us firstly select the two structured coclass trees (SCTs) T 2 (⟨243, 6⟩) and T 2 (⟨243, 8⟩) , which are known from Figures 8 and 9 already, and endow these trees with additional arithmetical structure by surrounding a realized vertex V with a circle and attaching an adjacent underlined boldface integer min{|d| | V = G23 (K(d))} which gives the minimal absolute discriminant such that V is realized by the second 3 -class group G23 (K(d)) . Then we obtain the arithmetically structured coclass trees (ASCTs) in Figures 10 and 11, which, in particular, give an impression of the actual distribution of second 3 -class groups. [5] See OEIS A242878 . Concerning the periodicity of occurrences of second 3 -class groups G23 (K(d)) of complex quadratic fields, it was proved [11] that only every other branch of the trees in Figures 10 and 11 can be populated by these metabelian 3 -groups and that the distribution sets in with a ground state (GS) on branch B(6) and continues with higher excited states (ES) on the branches B(j) with even j ≥ 8 . This periodicity phenomenon is underpinned by three sequences with fixed TKTs [10] • E.14 κ = (3122) , OEIS A247693 , • E.6 κ = (1122) , OEIS A247692 , • H.4 κ = (2122) , OEIS A247694 on the ASCT T 2 (⟨243, 6⟩) , and by three sequences with fixed TKTs [10] • E.9 κ = (2334) , OEIS A247696 , • E.8 κ = (2234) , OEIS A247695 , • G.16 κ = (2134) ,OEIS A247697
23.14. ARITHMETICAL APPLICATIONS
91
Figure 11: Minimal discriminants for the second ASCT of 3-groups with coclass 2 and abelianization (3,3).
on the ASCT T 2 (⟨243, 8⟩) . Up to now, [16] the ground state and three excited states are known for each of the six sequences, and for TKT E.9 κ = (2334) even the fourth excited state occurred already. The minimal absolute discriminants of the various states of each of the six periodic sequences are presented in Table 2. Data for the ground states (GS) and the first excited states (ES1) has been taken from D. C. Mayer, [11] most recent information on the second, third and fourth excited states (ES2, ES3, ES4) is due to N. Boston, M. R. Bush and F. Hajir. [16] In contrast, let us secondly select the sporadic part G0 (3, 2) of the coclass graph G(3, 2) for demonstrating that another way of attaching additional arithmetical structure to descendant trees is to display the counter #{|d| < b | V = G23 (K(d))} of hits of a realized vertex V by the second 3 -class group G23 (K(d)) of fields with absolute
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Figure 12: Frequency of sporadic 3-groups with coclass 2 and abelianization (3,3).
discriminants below a given upper bound b , for instance b = 108 . With respect to the total counter 276 375 of all complex quadratic fields with 3 -class group of type (3, 3) and discriminant −b < d < 0 , this gives the relative frequency as an approximation to the asymptotic density of the population in Figure 12 and Table 3. Exactly four vertices of the finite sporadic part G0 (3, 2) of G(3, 2) are populated by second 3 -class groups G23 (K(d)) : • ⟨243, 5⟩ , OEIS A247689 , • ⟨243, 7⟩ , OEIS A247690 , • ⟨729, 45⟩ , OEIS A242873 , • ⟨729, 57⟩ , OEIS A247688 .
23.14. ARITHMETICAL APPLICATIONS
93
Figure 13: Minimal absolute discriminants of sporadic 3-groups with coclass 2 and abelianization (3,3).
23.14.2
Comparison of various primes
√ Now let p ∈ {3, 5, 7} and consider complex quadratic fields K(d) = Q( d) with fixed signature (0, 1) and p-class groups of type (p, p) . The dominant part of the second p-class groups of these fields populates the top vertices of order p5 of the sporadic part G0 (p, 2) of the coclass graph G(p, 2) , which belong to the stem of P. Hall’s isoclinism family Φ6 , or their immediate descendants of order p6 . For primes p > 3 , the stem of Φ6 consists of p + 7 regular p-groups and reveals a rather uniform behaviour with respect to TKTs and TTTs, but the seven 3 -groups in the stem of Φ6 are irregular. We emphasize that there also exist several ( 3 for p = 3 and 4 for p > 3 ) infinitely capable vertices in the stem of Φ6 which are partially roots of coclass trees. However, here we focus on the sporadic vertices which are either isolated Schur σ -groups ( 2 for p = 3 and p + 1 for p > 3 ) or roots of finite trees within G0 (p, 2) ( 2 for each p ≥ 3 ). For p > 3 , the TKT of Schur σ -groups is a permutation whose cycle decomposition does not contain transpositions, whereas the TKT of roots of finite trees is a compositum of disjoint transpositions having an even number ( 0 or 2 ) of fixed points. We endow the forest G0 (p, 2) (a finite union of descendant trees) with additional arithmetical structure by attaching the minimal absolute discriminant min{|d| | V = G2p (K(d))} to each realized vertex V ∈ G0 (p, 2) . The resulting
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Figure 14: Minimal absolute discriminants of sporadic 5-groups with coclass 2 and abelianization (5,5).
structured sporadic coclass graph is shown in Figure 13 for p = 3 , in Figure 14 for p = 5 , and in Figure 15 for p=7.
23.15 References [1] Huppert, B. (1979). Endliche Gruppen I. Grundlehren der mathematischen Wissenschaften, Vol. 134, Springer-Verlag Berlin Heidelberg New York. [2] Schur, I. (1902). “Neuer Beweis eines Satzes über endliche Gruppen”. Sitzungsb. Preuss. Akad. Wiss.: 1013–1019. [3] Artin, E. (1929). “Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz”. Abh. Math. Sem. Univ. Hamburg 7: 46–51. [4] Blackburn, N. (1958). “On a special class of p-groups”. Acta Math. 100: 45–92. doi:10.1007/bf02559602. [5] Mayer, D. C. (2013). “The distribution of second p-class groups on coclass graphs”. J. Théor. Nombres Bordeaux 25 (2): 401–456. doi:10.5802/jtnb.842. [6] Chang, S. M., Foote, R. (1980). “Capitulation in class field extensions of type (p,p)". Can. J. Math. 32 (5): 1229–1243. doi:10.4153/cjm-1980-091-9. [7] Besche, H. U., Eick, B., O'Brien, E. A. (2005). The SmallGroups Library – a library of groups of small order. An accepted and refereed GAP 4 package, available also in MAGMA. [8] Besche, H. U., Eick, B., O'Brien, E. A. (2002). “A millennium project: constructing small groups”. Int. J. Algebra Comput. 12: 623–644. doi:10.1142/s0218196702001115.
23.15. REFERENCES
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Figure 15: Minimal absolute discriminants of sporadic 7-groups with coclass 2 and abelianization (7,7).
[9] Bush, M. R., Mayer, D. C. (2015). “3-class field towers of exact length 3”. J. Number Theory 147: 766–777 (preprint: arXiv:1312.0251 [math.NT], 2013). doi:10.1016/j.jnt.2014.08.010. [10] Mayer, D. C. (2012). “Transfers of metabelian p-groups”. Monatsh. Math. 166 (3-4): 467–495. doi:10.1007/s00605010-0277-x. [11] Mayer, D. C. (2012). “The second p-class group of a number field”. Int. J. Number Theory 8 (2): 471–505. doi:10.1142/s179304211250025x. [12] Scholz, A., Taussky, O. (1934). “Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm”. J. Reine Angew. Math. 171: 19–41. [13] Newman, M. F. (1977). Determination of groups of prime-power order. pp. 73-84, in: Group Theory, Canberra, 1975, Lecture Notes in Math., Vol. 573, Springer, Berlin. [14] O'Brien, E. A. (1990). “The p-group generation algorithm”. J. Symbolic Comput. 9: 677–698. doi:10.1016/s07477171(08)80082-x. [15] Miech, R. J. (1970). “Metabelian p-groups of maximal class”. Trans. Amer. Math. Soc. 152: 331–373. doi:10.1090/s00029947-1970-0276343-7. [16] Boston, N., Bush, M. R., Hajir, F. (2013). “Heuristics for p-class towers of imaginary quadratic fields”. Math. Ann. arXiv:1111.4679v1.
Chapter 24
Artin’s conjecture on primitive roots This page discusses a conjecture of Emil Artin on primitive roots. For the conjecture of Artin on L-functions, see Artin L-function. In number theory, Artin’s conjecture on primitive roots states that a given integer a which is neither a perfect square nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin’s constant or a rational multiple thereof. The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter’s diary. Although significant progress has been made, the conjecture is still unresolved as of May 2014. In fact, there is no single value of a for which Artin’s conjecture is proved.
24.1 Formulation Let a be an integer which is not a perfect square and not −1. Write a = a0 b2 with a0 square-free. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then 1. S(a) has a positive asymptotic density inside the set of primes. In particular, S(a) is infinite. 2. Under the conditions that a is not a perfect power and that a0 is not congruent to 1 modulo 4, this density is independent of a and equals Artin’s constant which can be expressed as an infinite product CArtin =
(
∏ p prime
1−
1 p(p−1)
)
= 0.3739558136 . . . (sequence A005596 in OEIS).
Similar conjectural product formulas [1] exist for the density when a does not satisfy the above conditions. In these cases, the conjectural density is always a rational multiple of CAᵣ ᵢ .
24.2 Example For example, take a = 2. The conjecture claims that the set of primes p for which 2 is a primitive root has the above density CAᵣ ᵢ . The set of such primes is (sequence A001122 in OEIS) S(2) = {3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, ...}. It has 38 elements smaller than 500 and there are 95 primes smaller than 500. The ratio (which conjecturally tends to CAᵣ ᵢ ) is 38/95 = 2/5 = 0.4. 96
24.3. PROOF ATTEMPTS
97
24.3 Proof attempts In 1967, Hooley published a conditional proof for the conjecture, assuming certain cases of the Generalized Riemann hypothesis.[2] In 1984, R. Gupta and M. Ram Murty showed unconditionally that Artin’s conjecture is true for infinitely many a using sieve methods.[3] Roger Heath-Brown improved on their result and showed unconditionally that there are at most two exceptional prime numbers a for which Artin’s conjecture fails.[4] This result is not constructive, as far as the exceptions go. For example, it follows from the theorem of Heath-Brown that one or more of 3, 5, and 7 is a primitive root modulo p for infinitely many p. But the proof does not provide us with a way of computing which one.
24.4 See also • The extensive review[5] by Pieter Moree • Brown–Zassenhaus conjecture • Full reptend prime • Cyclic number (group theory)
24.5 References [1] Michon, Gerard P. (2006-06-15). “Artin’s Constant”. Numericana. [2] Hooley, Christopher (1967). “On Artin’s conjecture”. J. Reine Angew. Math. 225: 209–220. doi:10.1515/crll.1967.225.209. MR 0207630. [3] Gupta, Rajiv; Murty, M. Ram (1984). “A remark on Artin’s conjecture”. Invent. Math. 78 (1): 127–130. doi:10.1007/BF01388719. MR 0762358. [4] Heath-Brown, David Rodney (1986). “Artin’s conjecture for primitive roots”. Quart. J. Math. Oxford Ser. 37 (1): 27–38. doi:10.1093/qmath/37.1.27. MR 0830627. [5] Moree, Pieter (2012). “Artin’s primitive root conjecture—a survey” (PDF). Integers 10 (6): 1305–1416. doi:10.1515/integers2012-0043. MR 3011564.
• Murty, M. Ram (1988). “Artin’s conjecture for primitive roots” (DVI). Mathematical Intelligencer 10 (4): 59–67. doi:10.1007/BF03023749. MR 0698163.
Chapter 25
Biquadratic field In mathematics, a biquadratic field is a number field K of a particular kind, which is a Galois extension of the rational number field Q with Galois group the Klein four-group. Such fields are all obtained by adjoining two square roots. Therefore in explicit terms we have K = Q(√a,√b) for rational numbers a and b. There is no loss of generality in taking a and b to be non-zero and square-free integers. According to Galois theory, there must be three quadratic fields contained in K, since the Galois group has three subgroups of index 2. The third subfield, to add to the evident Q(√a) and Q(√b), is Q(√ab). Biquadratic fields are the simplest examples of abelian extensions of Q that are not cyclic extensions. According to general theory the Dedekind zeta-function of such a field is a product of the Riemann zeta-function and three Dirichlet L-functions. Those L-functions are for the Dirichlet characters which are the Jacobi symbols attached to the three quadratic fields. Therefore taking the product of the Dedekind zeta-functions of the quadratic fields, multiplying them together, and dividing by the square of the Riemann zeta-function, is a recipe for the Dedekind zeta-function of the biquadratic field. This illustrates also some general principles on abelian extensions, such as the calculation of the conductor of a field. Such L-functions have applications in analytic theory (Siegel zeroes), and in some of Kronecker's work.
25.1 References • Section 12 of Swinnerton-Dyer, H.P.F. (2001), A brief guide to algebraic number theory, London Mathematical Society Student Texts 50, Cambridge University Press, ISBN 978-0-521-00423-7, MR 1826558
98
Chapter 26
Brauer group In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras.
26.1 Construction A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A, which is a simple ring, and for which the center is exactly K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – M(n,R) or M(n,H) – is a CSA over the reals, but not a division algebra (if n > 1 ). We obtain an equivalence relation on CSAs over K by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n,D) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying M(m,D) with M(n,D) for all integers m and n at least 1, we get the Brauer equivalence and the Brauer classes. Given central simple algebras A and B, one can look at the their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K. Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra Aop (the opposite ring with the same action by K since the image of K → A is in the center of A). In other words, for a CSA A we have A ⊗ Aop = M(n2 ,K), where n is the degree of A over K. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.)
26.2 Examples • In the following cases, every finite-dimensional central division algebra over a field K is K itself, so that the Brauer group Br(K) is trivial: •
• K is an algebraically closed field:[1] more generally, this is true for any pseudo algebraically closed field[2] or quasi-algebraically closed field.[3]
•
• K is a finite field (Wedderburn’s theorem);[1][4] 99
100
CHAPTER 26. BRAUER GROUP
•
• K is the function field of an algebraic curve over an algebraically closed field (Tsen’s theorem);[4]
•
• An algebraic extension of Q containing all roots of unity.[4]
• The Brauer group Br(R) of the field R of real numbers is the cyclic group of order two. There are just two non-isomorphic real division algebras with center R: the algebra R itself and the quaternion algebra H.[5] Since H ⊗ H ≅ M(4,R), the class of H has order two in the Brauer group. More generally, any real closed field has Brauer group of order two.[1] • K is complete under a discrete valuation with finite residue field. Br(K) is isomorphic to Q/Z.[5]
26.3 Brauer group and class field theory The notion of Brauer group plays an important role in the modern formulation of the class field theory. If Kv is a non-archimedean local field, the Hasse invariants gives a canonical isomorphism invv: Br(Kv) → Q/Z constructed in local class field theory.[6][7][8] An element of the Brauer group of order n can be represented by a cyclic division algebra of dimension n2 .[9] The case of a global field K is addressed by the global class field theory. If D is a central simple algebra over K and v is a valuation then D ⊗ Kv is a central simple algebra over Kv, the local completion of K at v. This defines a homomorphism from the Brauer group of K into the Brauer group of Kv. A given central simple algebra D splits for all but finitely many v, so that the image of D under almost all such homomorphisms is 0. The Brauer group Br(K) fits into an exact sequence[5][10] 0 → Br(K) →
⊕
Br(Kv ) → Q/Z → 0,
v∈S
where S is the set of all valuations of K and the right arrow is the direct sum of the local invariants: the Brauer group of the real numbers is identified with (1/2)Z/Z. The injectivity of the left arrow is the content of the Albert–Brauer– Hasse–Noether theorem. Exactness in the middle term is a deep fact from the global class field theory. The group Q/Z on the right may be interpreted as the “Brauer group” of the class formation of idele classes associated to K.
26.4 Properties • Base change from a field K to an extension field L gives a restriction map from Br(K) to Br(L). The kernel is the group Br(L/K) of classes of K-algebras that split over L. • The Brauer group of any field is a torsion group.[11]
26.5 General theory For an arbitrary field K, the Brauer group may be expressed in terms of Galois cohomology as follows:[12] ∗ Br(K) ∼ = H 2 (Gal(K s /K), K s ).
Here, K s is the separable closure of K, which coincides with the algebraic closure when K is a perfect field. Note that every finite dimensional central simple algebra has a separable splitting field.[13] The isomorphism of the Brauer group with a Galois cohomology group can be described as follows. If D is a division algebra over K of dimension n2 containing a Galois extension L of degree n over K, then the subgroup of elements of D* that normalize L is an extension of the Galois group Gal(L/K) by the nonzero elements L* of L, so corresponds to an element of H2 (Gal(L/K), L*). A generalisation of the Brauer group to the case of commutative rings was introduced by Maurice Auslander and Oscar Goldman,[14] and more generally for schemes by Alexander Grothendieck. In their approach, central simple algebras over a field are replaced with Azumaya algebras.[15]
26.6. SEE ALSO
101
26.6 See also • Algebraic K-theory
26.7 Notes [1] Lorenz (2008) p.164 [2] Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.). Springer-Verlag. p. 209. ISBN 978-3-540-77269-9. Zbl 1145.12001. [3] Serre (1979) p.161 [4] Serre (1979) p.162 [5] Serre (1979) p.163 [6] Lorenz (2008) p.232 [7] Serre (1967) p.137 [8] Shatz (1972) p.155 [9] Lorenz (2008) p.226 [10] Gille & Szamuely (2006) p.159 [11] Lorenz (2008) p.194 [12] Serre (1979) pp.157-159 [13] Jacobson (1996) p.93 [14] Auslander, Maurice; Goldman, Oscar (1961). “The Brauer group of a commutative ring”. Trans. Am. Math. Soc. 97: 367–409. doi:10.1090/s0002-9947-1960-0121392-6. ISSN 0002-9947. Zbl 0100.26304. [15] Saltman (1999) p.21
26.8 References • V.A. Iskovskikh (2001), “Brauer group of a field k", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001. • Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001. • Jacobson, Nathan (1996). Finite-dimensional division algebras over fields. Berlin: Springer-Verlag. ISBN 3-540-57029-2. Zbl 0874.16002. • Pierce, Richard (1982). Associative algebras. Graduate Texts in Mathematics 88. New York-Berlin: SpringerVerlag. ISBN 0-387-90693-2. Zbl 0497.16001. • Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series 28. Oxford University Press. pp. 237–241. ISBN 0-19-852673-3. Zbl 1024.16008. • Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.
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• Serre, Jean-Pierre (1967). “VI. Local class field theory”. In Cassels, J.W.S.; Fröhlich, A. Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. London: Academic Press. pp. 128–161. Zbl 0153.07403. • Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics 67. Translated from the French by Marvin Jay Greenberg. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016. • Shatz, Stephen S. (1972). Profinite groups, arithmetic, and geometry. Annals of Mathematics Studies 67. Princeton, NJ: Princeton University Press. ISBN 0-691-08017-8. MR 0347778. Zbl 0236.12002.
26.9 Further reading • DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics 181. Berlin-Heidelberg-New York: Springer-Verlag. ISBN 978-3-540-05371-2. Zbl 0215.36602.
26.10 External links • PlanetMath page • MathWorld page
Chapter 27
Brauer–Wall group In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field F is a group BW(F) classifying finite-dimensional graded central division algebras over the field. It was first defined by Terry Wall (1964) as a generalization of the Brauer group. The Brauer group of a field F is the set of the similarity classes of finite dimensional central simple algebras over F under the operation of tensor product, where two algebras are called similar if the commutants of their simple modules are isomorphic. Every similarity class contains a unique division algebra, so the elements of the Brauer group can also be identified with isomorphism classes of finite dimensional central division algebras. The analogous construction for Z/2Z-graded algebras defines the Brauer–Wall group BW(F).[1]
27.1 Properties • The Brauer group B(F) injects into BW(F) by mapping a CSA A to the graded algebra which is A in grade zero. • Wall (1964, theorem 3) showed that there is an exact sequence 0 → B(F) → BW(F) → Q(F) → 0 where Q(F) is the group of graded quadratic extensions of F, defined as an extension of Z/2 by F * /F *2 with multiplication (e,x)(f,y) = (e + f, (−1)ef xy). The map from W to BW is the Clifford invariant defined by mapping an algebra to the pair consisting of its grade and determinant. • There is a map from the additive group of the Witt–Grothendieck ring to the Brauer–Wall group obtained by sending a quadratic space to its Clifford algebra. The map factors through the Witt group,[2] which has kernel I3 , where I is the fundamental ideal of W(F).[3]
27.2 Examples • BW(C) is isomorphic to Z/2Z. This is an algebraic aspect of Bott periodicity of period 2 for the unitary group. The 2 super division algebras are C, C[γ] where γ is an odd element of square 1 commuting with C. • BW(R) is isomorphic to Z/8Z. This is an algebraic aspect of Bott periodicity of period 8 for the orthogonal group. The 8 super division algebras are R, R[ε], C[ε], H[δ], H, H[ε], C[δ], R[δ] where δ and ε are odd elements of square –1 and 1, such that conjugation by them on complex numbers is complex conjugation.
27.3 Notes [1] Lam (2005) pp.98–99
103
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CHAPTER 27. BRAUER–WALL GROUP
[2] Lam (2005) p.113 [3] Lam (2005) p.115
27.4 References • Deligne, Pierre (1999), “Notes on spinors”, in Deligne, Pierre; Etingof, Pavel; Freed, Daniel S.; Jeffrey, Lisa C.; Kazhdan, David; Morgan, John W.; Morrison, David R.; Witten, Edward, Quantum fields and strings: a course for mathematicians, Vol. 1, Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997, Providence, R.I.: American Mathematical Society, pp. 99–135, ISBN 978-0-8218-1198-6, MR 1701598 • Lam, Tsit-Yuen (2005), Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society, ISBN 0-8218-1095-2, MR 2104929, Zbl 1068.11023 • Wall, C. T. C. (1964), “Graded Brauer groups”, Journal für die reine und angewandte Mathematik 213: 187– 199, ISSN 0075-4102, MR 0167498, Zbl 0125.01904
Chapter 28
Brumer–Stark conjecture The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger’s theorem about the factorization of Gauss sums. It is named after Armand Brumer and Harold Stark. It arises as a special case (abelian and first-order) of Stark’s conjecture, when the place that splits completely in the extension is finite. There are very few cases where the conjecture is known to be valid. Its importance arises, for instance, from its connection with Hilbert’s twelfth problem.
28.1 Statement of the conjecture Let K/k be an abelian extension of global fields, and let S be a set of places of k containing the Archimedean places and the prime ideals that ramify in K/k. The S-imprimitive equivariant Artin L-function θ(s) is obtained from the usual equivariant Artin L-function by removing the Euler factors corresponding to the primes in S from the Artin L-functions from which the equivariant function is built. It is a function on the complex numbers taking values in the complex group ring C[G] where G is the Galois group of K/k. It is analytic on the entire plane, excepting a lone simple pole at s = 1. Let μK be the group of roots of unity in K. The group G acts on μK; let A be the annihilator of μK as a Z[G]-module. An important theorem, first proved by C. L. Siegel and later independently by Takuro Shintani, states that θ(0) is actually in Q[G]. A deeper theorem, proved independently by Pierre Deligne and Ken Ribet, Daniel Barsky, and Pierrette Cassou-Noguès, states that Aθ(0) is in Z[G]. In particular, Wθ(0) is in Z[G], where W is the cardinality of μK. The ideal class group of K is a G-module. From the above discussion, we can let Wθ(0) act on it. The Brumer–Stark conjecture says the following:[1] Brumer–Stark Conjecture. For each nonzero fractional ideal a of K, there is an “anti-unit” ε such that 1. aW θ(0) = (ε).
( 1) 2. The extension K ε W /k is abelian. The first part of this conjecture is due to Armand Brumer, and Harold Stark originally suggested that the second condition might hold. The conjecture was first stated in published form by John Tate.[2] The term “anti-unit” refers to the condition that |ε|ν is required to be 1 for each Archimedean place ν.[1]
28.2 Progress The Brumer Stark conjecture is known to be true for extensions K/k where • K/Q is cyclotomic: this follows from Stickelberger’s theorem[1] 105
106
CHAPTER 28. BRUMER–STARK CONJECTURE
• k is abelian over Q[3] • K/k is a quadratic extension[2] • K/k is a biquadratic extension[4]
28.3 Function field analogue The analogous statement in the function field case is known to be true, having been proved by John Tate and Pierre Deligne, with a different proof by David Hayes.[5]
28.4 References [1] Lemmermeyer, Franz (2000). Reciprocity laws. From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer-Verlag. p. 384. ISBN 3-540-66957-4. MR 1761696. Zbl 0949.11002. [2] Tate, John, Brumer–Stark–Stickelberger, Séminaire de Théorie des Nombres, Univ. Bordeaux I Talence, (1980-81), exposé no. 24. [3] Tate, John, “Les Conjectures de Stark sur les Fonctions L d'Artin en s=0”, Progress in Mathematics (Birkhauser) 47, MR 86e:11112 [4] Sands, J. W. (1984), “Galois groups of exponent 2 and the Brumer–Stark conjecture”, J. Reine Angew. Math 349 (1): 129–135, doi:10.1515/crll.1984.349.129 [5] Rosen, Michael (2002), “15. The Brumer-Stark conjecture”, Number theory in function fields, Graduate Texts in Mathematics 210, New York, NY: Springer-Verlag, ISBN 0-387-95335-3, Zbl 1043.11079
Chapter 29
Carlitz exponential In mathematics, the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real and complex analysis. It is used in the definition of the Carlitz module – an example of a Drinfeld module.
29.1 Definition We work over the polynomial ring Fq[T] of one variable over a finite field Fq with q elements. The completion C∞ of an algebraic closure of the field Fq((T −1 )) of formal Laurent series in T −1 will be useful. It is a complete and algebraically closed field. First we need analogues to the factorials, which appear in the definition of the usual exponential function. For i > 0 we define
i
[i] := T q − T, Di :=
∏
[j]q
i−j
1≤j≤i
and D0 := 1. Note that that the usual factorial is inappropriate here, since n! vanishes in Fq[T] unless n is smaller than the characteristic of Fq[T]. Using this we define the Carlitz exponential eC:C∞ → C∞ by the convergent sum
eC (x) :=
j ∞ ∑ xq
j=0
Di
.
29.2 Relation to the Carlitz module The Carlitz exponential satisfies the functional equation
q
eC (T x) = T eC (x) + (eC (x)) = (T + τ )eC (x), where we may view τ as the power of q map or as an element of the ring Fq (T ){τ } of noncommutative polynomials. By the universal property of polynomial rings in one variable this extends to a ring homomorphism ψ:Fq[T]→C∞{τ}, defining a Drinfeld Fq[T]-module over C∞{τ}. It is called the Carlitz module. 107
108
CHAPTER 29. CARLITZ EXPONENTIAL
29.3 References • Goss, D. (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 35, Berlin, New York: Springer-Verlag, ISBN 978-3-540-61087-8, MR 1423131 • Thakur, Dinesh (2004), Function field arithmetic, New Jersey: World Scientific Publishing, ISBN 981-238839-7, MR 2091265 |first2= missing |last2= in Authors list (help)
Chapter 30
Characteristic (algebra) In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring’s multiplicative identity element (1) in a sum to get the additive identity element (0); the ring is said to have characteristic zero if this sum never reaches the additive identity. That is, char(R) is the smallest positive number n such that
1 + ··· + 1 = 0 | {z } nsummands
if such a number n exists, and 0 otherwise. The characteristic may also be taken to be the exponent of the ring’s additive group, that is, the smallest positive n such that
a + ··· + a = 0 | {z } nsummands
for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see ring), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.
30.1 Other equivalent characterizations • The characteristic is the natural number n such that nZ is the kernel of a ring homomorphism from Z to R; • The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z/nZ, which would be the image of that homomorphism. • When the non-negative integers {0, 1, 2, 3, . . . } are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of n for which n · 1 = 0. If nothing “smaller” (in this ordering) than 0 will suffice, then the characteristic is 0. This is the right partial ordering because of such facts as that char A × B is the least common multiple of char A and char B, and that no ring homomorphism ƒ : A → B exists unless char B divides char A. • The characteristic of a ring R is n ∈ {0, 1, 2, 3, . . . } precisely if the statement ka = 0 for all a ∈ R implies n is a divisor of k. The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring of integers to any ring; in the language of category theory, Z is an initial object of the category of rings. Again this follows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms). 109
110
CHAPTER 30. CHARACTERISTIC (ALGEBRA)
30.2 Case of rings If R and S are rings and there exists a ring homomorphism R → S, then the characteristic of S divides the characteristic of R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the trivial ring which has only a single element 0 = 1. If a non-trivial ring R does not have any zero divisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic 0 is infinite. The ring Z/nZ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same characteristic. For instance, if q(X) is a prime polynomial with coefficients in the field Z/pZ where p is prime, then the factor ring (Z/pZ)[X]/(q(X)) is a field of characteristic p. Since the complex numbers contain the rationals, their characteristic is 0. A Z/nZ-algebra is equivalently a ring whose characteristic divides n. If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R – the "freshman’s dream" holds for power p. The map f(x) = xp then defines a ring homomorphism R → R. It is called the Frobenius homomorphism. If R is an integral domain it is injective.
30.3 Case of fields As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of finite characteristic or a field of positive characteristic. For any field F, there is a minimal subfield, namely the prime field, the smallest subfield containing 1F. It is isomorphic either to the rational number field Q, or a finite field of prime order, Fp; the structure of the prime field and the characteristic each determine the other. Fields of characteristic zero have the most familiar properties; for practical purposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is; in fact, any field of characteristic zero and cardinality at most continuum is isomorphic to a subfield of complex numbers).[1] The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic pk , as k → ∞. For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus, number fields and the field of complex numbers C are of characteristic zero. Actually, every field of characteristic zero is the quotient field of a ring Q[X]/P where X is a set of variables and P a set of polynomials in Q[X]. The finite field GF(pn ) has characteristic p. There exist infinite fields of prime characteristic. For example, the field of all rational functions over Z/pZ, the algebraic closure of Z/pZ or the field of formal Laurent series Z/pZ((T)). The characteristic exponent is defined similarly, except that it is equal to 1 if the characteristic is zero; otherwise it has the same value as the characteristic.[2] The size of any finite ring of prime characteristic p is a power of p. Since in that case it must contain Z/pZ it must also be a vector space over that field and from linear algebra we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size pn . So its size is (pn )m = pnm .)
30.4 References [1] Enderton, Herbert B. (2001), A Mathematical Introduction to Logic (2nd ed.), Academic Press, p. 158, ISBN 9780080496467. Enderton states this result explicitly only for algebraically closed fields, but also describes a decomposition of any field as an algebraic extension of a transcendental extension of its prime field, from which the result follows immediately.
30.4. REFERENCES
[2] “Field Characteristic Exponent”. Wolfram Mathworld. Wolfram Research. Retrieved May 27, 2015.
• Neal H. McCoy (1964, 1973) The Theory of Rings, Chelsea Publishing, page 4.
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Chapter 31
Class field theory
+i
−1
0
+1
−i
The fifth roots of unity in the complex plane. Adding these roots to the rational numbers generates an abelian extension.
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields and function fields of curves over finite fields and arithmetic properties of such abelian extensions. A general name for such fields is global fields, or one-dimensional global fields. 112
31.1. FORMULATION IN CONTEMPORARY LANGUAGE
113
The theory takes its name from the fact that it provides a one-to-one correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field. For example, the Hilbert class field, which is the maximal unramified abelian extension of a number field, corresponds to a very special class of ideals. Class field theory also includes a reciprocity homomorphism, which acts from the idele class group of a global field, i.e. the quotient of the ideles by the multiplicative group of the field, to the Galois group of the maximal abelian extension of the global field. Each open subgroup of the idele class group of a global field is the image with respect to the norm map from the corresponding class field extension down to the global field. A standard method since the 1930s is to develop local class field theory, which describes abelian extensions of completions of a global field, and then use it to construct global class field theory.
31.1 Formulation in contemporary language In modern language there is a maximal abelian extension A of K, which will be of infinite degree over K; and associated to A a Galois group G, which will be a pro-finite group, so a compact topological group, and also abelian. The central aim of the theory is to describe G in terms of K. In particular to establish a one-to-one correspondence between finite abelian extensions of K and their norm groups in an appropriate object for K, such as the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields, as well as to describe those norm groups directly, e.g., such as open subgroups of finite index. The finite abelian extension corresponding to such a subgroup is called a class field, which gave the name to the theory. The fundamental result of class field theory states that the group G is naturally isomorphic to the profinite completion of the idele class group CK of K with respect to the natural topology on CK related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism Gal(L / K)ab → CK / NL/K CL of the maximal abelian quotient of the Galois group of the extension with the quotient of the idele class group of K by the image of the norm of the idele class group of L.[1] For some small fields, such as the field of rational numbers Q or its quadratic imaginary extensions there is a more detailed theory which provides more information. For example, the abelianized absolute Galois group G of Q is (naturally isomorphic to) an infinite product of the group of units of the p-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity. This is known as the Kronecker–Weber theorem, originally conjectured by Leopold Kronecker. In this case the reciprocity isomorphism of class field theory (or Artin reciprocity map) also admits an explicit description due to the Kronecker– Weber theorem. Let us denote with
µ∞ ⊂ C× the group of all roots of unity, i.e. the torsion subgroup. The Artin reciprocity map is given by
ˆ × → Gab = Gal(Q(µ∞ )/Q), Z Q
x 7→ (ζ 7→ ζ x ),
when it is arithmetically normalized, or given by
ˆ × → Gab = Gal(Q(µ∞ )/Q), Z Q
x 7→ (ζ 7→ ζ −x ),
if it is geometrically normalized. However, principal constructions of such more detailed theories for small algebraic number fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory. The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension (this is done inside local class field theory) and then prove that the product of all such local reciprocity maps when
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defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field. The latter property is called the global reciprocity law and is a far reaching generalization of the Gauss quadratic reciprocity law. One of the methods to construct the reciprocity homomorphism uses class formation. There are methods which use cohomology groups, in particular the Brauer group, and there are methods which do not use cohomology groups and are very explicit and good for applications.
31.2 Prime ideals More than just the abstract description of G, it is essential for the purposes of number theory to understand how prime ideals decompose in the abelian extensions. The description is in terms of Frobenius elements, and generalises in a far-reaching way the quadratic reciprocity law that gives full information on the decomposition of prime numbers in quadratic fields. The class field theory project included the 'higher reciprocity laws’ (cubic reciprocity) and so on.
31.3 Generalizations of class field theory One natural development in number theory is to understand and construct nonabelian class field theories which provide information about general Galois extensions of global fields. Often, the Langlands correspondence is viewed as a nonabelian class field theory and indeed when fully established it will contain a very rich theory of nonabelian Galois extensions of global fields. However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case. It also does not include an analog of the existence theorem in class field theory, i.e. the concept of class fields is absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternative to the Langlands correspondence point of view. Another natural development in arithmetic geometry is to understand and construct class field theory which describes abelian extensions of higher local and global fields. The latter come as function fields of schemes of finite type over integers and their appropriate localization and completions. Higher local and global class field theory uses algebraic K-theory and appropriate Milnor K-groups replace K1 which is in use in one-dimensional class field theory. Higher local and global class field theory was developed by A. Parshin, Kazuya Kato, Ivan Fesenko, Spencer Bloch, Shuji Saito and other mathematicians. There are attempts to develop higher global class field theory without using algebraic K-theory (G. Wiesend), but his approach does not involve higher local class field theory and a compatibility between the local and global theories.
31.4 History Main article: History of class field theory The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalisation took place as a long-term historical project, involving quadratic forms and their 'genus theory', work of Ernst Kummer and Leopold Kronecker/Kurt Hensel on ideals and completions, the theory of cyclotomic and Kummer extensions. The first two class field theories were very explicit cyclotomic and complex multiplication class field theories. They used additional structures: in the case of the field of rational numbers they use roots of unity, in the case of imaginary quadratic extensions of the field of rational numbers they use elliptic curves with complex multiplication and their points of finite order. Much later, the theory of Shimura provided another very explicit class field theory for a class of algebraic number fields. All these very explicit theories cannot be extended to work over arbitrary number field. In positive characteristic p Kawada and Satake used Witt duality to get a very easy description of the p -part of the reciprocity homomorphism. However, general class field theory used different concepts and its constructions work over every global field. The famous problems of David Hilbert stimulated further development, which lead to the reciprocity laws, and proofs by Teiji Takagi, Phillip Furtwängler, Emil Artin, Helmut Hasse and many others. The crucial Takagi existence theorem was known by 1920 and all the main results by about 1930. One of the last classical conjectures to be
31.5. REFERENCES
115
proved was the principalisation property. The first proofs of class field theory used substantial analytic methods. In the 1930s and subsequently the use of infinite extensions and the theory of Wolfgang Krull of their Galois groups was found increasingly useful. It combines with Pontryagin duality to give a clearer if more abstract formulation of the central result, the Artin reciprocity law. An important step was the introduction of ideles by Claude Chevalley in 1930s. Their use replaced the classes of ideals and essentially clarified and simplified structures that describe abelian extensions of global fields. Most of the central results were proved by 1940. Later the results were reformulated in terms of group cohomology, which became a standard way to learn class field theory for several generations of number theorists. One drawback of the cohomological method is its relative inexplicitness. As the result of local contributions by Bernard Dwork, John Tate, Michiel Hazewinkel and a local and global reinterpretation by Jürgen Neukirch and also in relation to the work on explicit reciprocity formulas by many mathematicians, a very explicit and cohomology free presentation of class field theory was established in the nineties, see e.g. the book of Neukirch.
31.5 References [1] (Neukirch 1999, Theorems VI.5.5, VI.6.1)
• Artin, Emil; Tate, John (1990), Class field theory, Redwood City, Calif.: Addison-Wesley, ISBN 978-0-20151011-9 • Cohen, Henri; Stevenhagen, Peter (2008), “Computational class field theory”, in Buhler, J.P.; P., Stevenhagen, Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, MSRI Publications 44, Cambridge University Press, pp. 497–534, ISBN 978-0-521-20833-8, Zbl 1177.11095 • Cassels, J.W.S.; Fröhlich, Albrecht, eds. (1967), Algebraic Number Theory, Academic Press, Zbl 0153.07403 • Iwasawa, Kenkichi (1986), Local class field theory, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-504030-2, MR 863740, Zbl 0604.12014 • Neukirch, Jürgen (1986), Class Field Theory, Berlin, New York: Springer-Verlag, ISBN 978-3-540-15251-4 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Kawada, Y. (1955), “Class formations”, Duke Math.J. 22: 165–177, Zbl 0067.01904 • Kawada, Y.; Satake, I. (1956), “Class formations. II”, J.Fac. Sci.Univ. Tokyo Sect. 1A 7: 353–389, Zbl 0101.02902 • Conrad, Keith, History of class field theory. (PDF) • Gras, Georges (second corrected printing 2005), Class field theory: From theory to practice, New York: Springer Check date values in: |date= (help)
Chapter 32
Class formation In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory.
32.1 Definitions A formation is a topological group G together with a topological G-module A on which G acts continuously. A layer E/F of a formation is a pair of open subgroups E, F of G such that F is a finite index subgroup of E. It is called a normal layer if F is a normal subgroup of E, and a cyclic layer if in addition the quotient group is cyclic. If E is a subgroup of G, then AE is defined to be the elements of A fixed by E. We write H n (E/F) for the Tate cohomology group H n (E/F, AF ) whenever E/F is a normal layer. (Some authors think of E and F as fixed fields rather than subgroup of G, so write F/E instead of E/F.) In applications, G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure. A class formation is a formation such that for every normal layer E/F H 1 (E/F) is trivial, and H 2 (E/F) is cyclic of order |E/F|. In practice, these cyclic groups come provided with canonical generators uE/F ∈ H 2 (E/F), called fundamental classes, that are compatible with each other in the sense that the restriction (of cohomology classes) of a fundamental class is another fundamental class. Often the fundamental classes are considered to be part of the structure of a class formation. A formation that satisfies just the condition H 1 (E/F)=1 is sometimes called a field formation. For example, if G is any finite group acting on a field A, then this is a field formation by Hilbert’s theorem 90.
32.2 Examples of class formations The most important examples of class formations (arranged roughly in order of difficulty) are as follows: • Archimedean local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by complex conjugation. • Finite fields: The module A is the integers (with trivial G-action), and G is the absolute Galois group of a finite field, which is isomorphic to the profinite completion of the integers. 116
32.3. THE FIRST INEQUALITY
117
• Local class field theory of characteristic p>0: The module A is the separable algebraic closure of the field of formal Laurent series over a finite field, and G is the Galois group. • Non-archimedean local class field theory of characteristic 0: The module A is the algebraic closure of a field of p-adic numbers, and G is the Galois group. • Global class field theory of characteristic p>0: The module A is the union of the groups of idele classes of separable finite extensions of some function field over a finite field, and G is the Galois group. • Global class field theory of characteristic 0: The module A is the union of the groups of idele classes of algebraic number fields, and G is the Galois group of the rational numbers (or some algebraic number field) acting on A. It is easy to verify the class formation property for the finite field case and the archimedean local field case, but the remaining cases are more difficult. Most of the hard work of class field theory consists of proving that these are indeed class formations. This is done in several steps, as described in the sections below.
32.3 The first inequality The first inequality of class field theory states that |H 0 (E/F)| ≥ |E/F| for cyclic layers E/F. It is usually proved using properties of the Herbrand quotient, in the more precise form |H 0 (E/F)| = |E/F|×|H 1 (E/F)|. It is fairly straighforward to prove, because the Herbrand quotient is easy to work out, as it is multiplicative on short exact sequences, and is 1 for finite modules. Before about 1950, the first inequality was known as the second inequality, and vice versa.
32.4 The second inequality The second inequality of class field theory states that |H 0 (E/F)| ≤ |E/F| for all normal layers E/F. For local fields, this inequality follows easily from Hilbert’s theorem 90 together with the first inequality and some basic properties of group cohomology. The second inequality was first proved for global fields by Weber using properties of the L series of number fields, as follows. Suppose that the layer E/F corresponds to an extension k⊂K of global fields. By studying the Dedekind zeta function of K one shows that the degree 1 primes of K have Dirichlet density given by the order of the pole at s=1, which is 1 (When K is the rationals, this is essentially Euler’s proof that there are infinitely many primes using the pole at s=1 of the Riemann zeta function.) As each prime in k that is a norm is the product of deg(K/k)= |E/F| distinct degree 1 primes of K, this shows that the set of primes of k that are norms has density 1/|E/F|. On the other hand, by studying Dirichlet L-series of characters of the group H 0 (E/F), one shows that the Dirichlet density of primes of k representing the trivial element of this group has density 1/|H 0 (E/F)|. (This part of the proof is a generalization of Dirichlet’s proof that there are infinitely many primes in arithmetic progressions.) But a prime represents a trivial element of the group H 0 (E/F) if it is equal to a norm modulo principal ideals, so this set is at least as dense as the set of primes that are norms. So 1/|H 0 (E/F)| ≥ 1/|E/F|
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which is the second inequality. In 1940 Chevalley found a purely algebraic proof of the second inequality, but it is longer and harder than Weber’s original proof. Before about 1950, the second inequality was known as the first inequality; the name was changed because Chevalley’s algebraic proof of it uses the first inequality. Takagi defined a class field to be one where equality holds in the second inequality. By the Artin isomorphism below, H 0 (E/F) is isomorphic to the abelianization of E/F, so equality in the second inequality holds exactly for abelian extensions, and class fields are the same as abelian extensions. The first and second inequalities can be combined as follows. For cyclic layers, the two inequalities together prove that H 1 (E/F)|E/F| = H 0 (E/F) ≤ |E/F| so H 0 (E/F) = |E/F| and H 1 (E/F) = 1. Now a basic theorem about cohomology groups shows that since H 1 (E/F) = 1 for all cyclic layers, we have H 1 (E/F) = 1 for all normal layers (so in particular the formation is a field formation). This proof that H 1 (E/F) is always trivial is rather roundabout; no “direct” proof of it (whatever this means) for global fields is known. (For local fields the vanishing of H 1 (E/F) is just Hilbert’s theorem 90.) For cyclic group, H 0 is the same as H 2 , so H 2 (E/F) = |E/F| for all cyclic layers. Another theorem of group cohomology shows that since H 1 (E/F) = 1 for all normal layers and H 2 (E/F) ≤ |E/F| for all cyclic layers, we have H 2 (E/F)≤ |E/F| for all normal layers. (In fact, equality holds for all normal layers, but this takes more work; see the next section.)
32.5 The Brauer group The Brauer groups H 2 (E/*) of a class formation are defined to be the direct limit of the groups H 2 (E/F) as F runs over all open subgroups of E. An easy consequence of the vanishing of H 1 for all layers is that the groups H 2 (E/F) are all subgroups of the Brauer group. In local class field theory the Brauer groups are the same as Brauer groups of fields, but in global class field theory the Brauer group of the formation is not the Brauer group of the corresponding global field (though they are related). The next step is to prove that H 2 (E/F) is cyclic of order exactly |E/F|; the previous section shows that it has at most this order, so it is sufficient to find some element of order |E/F| in H 2 (E/F). The proof for arbitrary extensions uses a homomorphism from the group G onto the profinite completion of the integers with kernel G∞, or in other words a compatible sequence of homomorphisms of G onto the cyclic groups of order n for all n, with kernels Gn. These homomorphisms are constructed using cyclic cyclotomic extensions of fields; for finite fields they are given by the algebraic closure, for non-archimedean local fields they are given by the maximal unramified extensions, and for global fields they are slightly more complicated. As these extensions are given explicitly one can check that they have the property that H2 (G/Gn) is cyclic of order n, with a canonical generator. It follows from this that for any layer E, the group H2 (E/E∩G∞) is canonically isomorphic to Q/Z. This idea of using roots of unity was introduced by Chebotarev in his proof of Chebotarev’s density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem.
32.6. TATE’S THEOREM AND THE ARTIN MAP
119
For general layers E,F there is an exact sequence
0 → H 2 (E/F ) ∩ H 2 (E/E ∩ G∞ ) → H 2 (E/E ∩ G∞ ) → H 2 (F /F ∩ G∞ ) The last two groups in this sequence can both be identified with Q/Z and the map between them is then multiplication by |E/F|. So the first group is canonically isomorphic to Z/nZ. As H 2 (E/F) has order at most Z/nZ is must be equal to Z/nZ (and in particular is contained in the middle group)). This shows that the second cohomology group H 2 (E/F) of any layer is cyclic of order |E/F|, which completes the verification of the axioms of a class formation. With a little more care in the proofs, we get a canonical generator of H 2 (E/F), called the fundamental class. It follows from this that the Brauer group H 2 (E/*) is (canonically) isomorphic to the group Q/Z, except in the case of the archimedean local fields R and C when it has order 2 or 1.
32.6 Tate’s theorem and the Artin map Tate’s theorem in group cohomology is as follows. Suppose that A is a module over a finite group G and a is an element of H 2 (G,A), such that for every subgroup E of G • H 1 (E,A) is trivial, and • H 2 (E,A) is generated by Res(a) which has order E. Then cup product with a is an isomorphism • H n (G,Z) → H n+2 (G,A). If we apply the case n=−2 of Tate’s theorem to a class formation, we find that there is an isomorphism • H −2 (E/F,Z) → H 0 (E/F,AF ) for any normal layer E/F. The group H −2 (E/F,Z) is just the abelianization of E/F, and the group H 0 (E/F,AF ) is AE modulo the group of norms of AF . In other words we have an explicit description of the abelianization of the Galois group E/F in terms of AE . Taking the inverse of this isomorphism gives a homomorphism AE → abelianization of E/F, and taking the limit over all open subgroups F gives a homomorphism AE → abelianization of E, called the Artin map. The Artin map is not necessarily surjective, but has dense image. By the existence theorem below its kernel is the connected component of AE (for class field theory), which is trivial for class field theory of non-archimedean local fields and for function fields, but is non-trivial for archimedean local fields and number fields.
32.7 The Takagi existence theorem The main remaining theorem of class field theory is the Takagi existence theorem, which states that every finite index closed subgroup of the idele class group is the group of norms corresponding to some abelian extension. The classical way to prove this is to construct some extensions with small groups of norms, by first adding in lots of roots of unity, and then taking Kummer extensions and Artin–Schreier extensions. These extensions may be non-abelian (though
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they are extensions of abelian groups by abelian groups); however, this does not really matter, as the norm group of a non-abelian Galois extension is the same as that of its maximal abelian extension (this can be shown using what we already know about class fields). This gives enough (abelian) extensions to show that there is an abelian extension corresponding to any finite index subgroup of the idele class group. A consequence is that the kernel of the Artin map is the connected component of the identity of the idele class group, so that the abelianization of the Galois group of F is the profinite completion of the idele class group. For local class field theory, it is also possible to construct abelian extensions more explicitly using Lubin–Tate formal group laws. For global fields, the abelian extensions can be constructed explicitly in some cases: for example, the abelian extensions of the rationals can be constructed using roots of unity, and the abelian extensions of quadratic imaginary fields can be constructed using elliptic functions, but finding an analog of this for arbitrary global fields is an unsolved problem.
32.8 Weil group Main article: Weil group
This is not a Weyl group and has no connection with the Weil–Châtelet group or the Mordell–Weil group The Weil group of a class formation with fundamental classes uE/F ∈ H 2 (E/F, AF ) is a kind of modified Galois group, introduced by Weil (1951) and used in various formulations of class field theory, and in particular in the Langlands program. If E/F is a normal layer, then the Weil group U of E/F is the extension 1 → AF → U → E/F → 1 corresponding to the fundamental class uE/F in H 2 (E/F, AF ). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers G/F, for F an open subgroup of G. The reciprocity map of the class formation (G, A) induces an isomorphism from AG to the abelianization of the Weil group.
32.9 See also • Abelian extension • Artin L-function • Artin reciprocity • Class field theory • Complex multiplication • Galois cohomology • Hasse norm theorem • Herbrand quotient • Hilbert class field • Kronecker–Weber theorem • Local class field theory • Takagi existence theorem • Tate cohomology group
32.10. REFERENCES
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32.10 References • Artin, Emil; Tate, John (2009) [1952], Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, MR 0223335 • Kawada, Yukiyosi (1971), “Class formations”, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Providence, R.I.: American Mathematical Society, pp. 96–114 • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 554237, esp. chapter XI: Class formations • Tate, J. (1979), “Number theoretic background”, Automorphic forms, representations, and L-functions Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 0-8218-1435-4 • Weil, André (1951), “Sur la theorie du corps de classes”, Journal of the Mathematical Society of Japan 3: 1–35, doi:10.2969/jmsj/00310001, ISSN 0025-5645, MR 0044569, reprinted in volume I of his collected papers, ISBN 0-387-90330-5
Chapter 33
Class number formula In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function
33.1 General statement of the class number formula We start with the following data: • K is a number field. • [K : Q] = n = r1 + 2r2 , where r1 denotes the number of real embeddings of K, and 2r2 is the number of complex embeddings of K. • ζK(s) is the Dedekind zeta function of K. • hK is the class number, the number of elements in the ideal class group of K. • RegK is the regulator of K. • wK is the number of roots of unity contained in K. • DK is the discriminant of the extension K/Q. Then: Theorem (Class Number Formula). ζK(s) converges absolutely for Re(s) > 1 and extends to a meromorphic function defined for all complex s with only one simple pole at s = 1, with residue
lim (s − 1)ζK (s) =
s→1
2r1 · (2π)r2 · hK · RegK √ wK · |DK |
This is the most general “class number formula”. In particular cases, for example when K is a cyclotomic extension of Q, there are particular and more refined class number formulas.
33.2 Proof The idea of the proof of the class number formula is most easily seen when K = Q(i). In this case, the ring of integers in K is the Gaussian integers. An elementary manipulation shows that the residue of the Dedekind zeta function at s = 1 is the average of the coefficients of the Dirichlet series representation of the Dedekind zeta function. The n-th coefficient of the Dirichlet 122
33.3. DIRICHLET CLASS NUMBER FORMULA
123
series is essentially the number of representations of n as a sum of two squares of nonnegative integers. So one can compute the residue of the Dedekind zeta function at s = 1 by computing the average number of representations. As in the article on the Gauss circle problem, one can compute this by approximating the number of lattice points inside of a quarter circle centered at the origin, concluding that the residue is one quarter of pi. The proof when K is an arbitrary imaginary quadratic number field is very similar.[1] In the general case, by Dirichlet’s unit theorem, the group of units in the ring of integers of K is infinite. One can nevertheless reduce the computation of the residue to a lattice point counting problem using the classical theory of real and complex embeddings[2] and approximate the number of lattice points in a region by the volume of the region, to complete the proof.
33.3 Dirichlet class number formula Peter Gustav Lejeune Dirichlet published a proof of the class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms rather than classes of ideals. It appears that Gauss already knew this formula in 1801.[3] This exposition follows Davenport.[4] Let d be a fundamental discriminant, and write h(d) for the number of equivalence classes of quadratic forms with (d) discriminant d. Let χ = m be the Kronecker symbol. Then χ is a Dirichlet character. Write L(s, χ) for the Dirichlet L-series based on χ . For d > 0, let t > 0, u > 0 be the solution to the Pell equation t2 − du2 = 4 for which u is smallest, and write
ϵ=
√ 1 (t + u d). 2
√ (Then ε is either a fundamental unit of the real quadratic field Q( d) or the square of a fundamental unit.) For d < 0, write w for the number of automorphs of quadratic forms of discriminant d; that is, 2, w = 4, 6,
d < −4; d = −4; d = −3.
Then Dirichlet showed that √ w |d| L(1, χ), d < 0; h(d) = √2π d L(1, χ), d > 0. ln ϵ This is a special case of Theorem 1 above: for a quadratic field K, the Dedekind zeta function is just ζK (s) = ζ(s)L(s, χ) , and the residue is L(1, χ) . Dirichlet also showed that the L-series can be written in a finite form, which gives a finite form for the class number. Suppose χ is primitive with prime conductor q . Then ( ) m π ∑q−1 , q ≡ 3 mod 4; − 3/2 m=1 m q ( )q L(1, χ) = − 1 ∑q−1 m ln 2 sin mπ , q ≡ 1 mod 4. q q 1/2 m=1 q
33.4 Galois extensions of the rationals If K is a Galois extension of Q, the theory of Artin L-functions applies to ζK (s) . It has one factor of the Riemann zeta function, which has a pole of residue one, and the quotient is regular at s = 1. This means that the right-hand side of the class number formula can be equated to a left-hand side
124
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with ρ running over the classes of irreducible non-trivial complex linear representations of Gal(K/Q) of dimension dim(ρ). That is according to the standard decomposition of the regular representation.
33.5 Abelian extensions of the rationals This is the case of the above, with Gal(K/Q) an abelian group, in which all the ρ can be replaced by Dirichlet characters (via class field theory) for some modulus f called the conductor. Therefore all the L(1) values occur for Dirichlet L-functions, for which there is a classical formula, involving logarithms. By the Kronecker–Weber theorem, all the values required for an analytic class number formula occur already when the cyclotomic fields are considered. In that case there is a further formulation possible, as shown by Kummer. The regulator, a calculation of volume in 'logarithmic space' as divided by the logarithms of the units of the cyclotomic field, can be set against the quantities from the L(1) recognisable as logarithms of cyclotomic units. There result formulae stating that the class number is determined by the index of the cyclotomic units in the whole group of units. In Iwasawa theory, these ideas are further combined with Stickelberger’s theorem.
33.6 Notes [1] https://www.math.umass.edu/~{}weston/oldpapers/cnf.pdf [2] http://planetmath.org/realandcomplexembeddings [3] http://mathoverflow.net/questions/109330/did-gauss-know-dirichlets-class-number-formula-in-1801 [4] Davenport, Harold (2000). Montgomery, Hugh L., ed. Multiplicative Number Theory. Graduate Texts in Mathematics 74 (3rd ed.). New York: Springer-Verlag. pp. 43–53. ISBN 978-0-387-95097-6. Retrieved 2009-05-26.
33.7 References • W. Narkiewicz (1990). Elementary and analytic theory of algebraic numbers (2nd ed ed.). Springer-Verlag/Polish Scientific Publishers PWN. pp. 324–355. ISBN 3-540-51250-0. This article incorporates material from Class number formula on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Chapter 34
Class number problem In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields with class number n. It is named after the great mathematician Carl Friedrich Gauss. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and the behavior as d → −∞ . The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number (meaning that they involve a constant that is not computed), but effective bounds (and explicit proofs of completeness of lists) are harder.
34.1 Gauss’s original conjectures The problems are posed in Gauss’s Disquisitiones Arithmeticae of 1801 (Section V, Articles 303 and 304).[1] Gauss discusses imaginary quadratic fields in Article 303, stating the first two conjectures, and discusses real quadratic fields in Article 304, stating the third conjecture. Gauss Conjecture (Class number tends to infinity) h(d) → ∞ as d → −∞. Gauss Class Number Problem (Low class number lists) For given low class number (such as 1, 2, and 3), Gauss gives lists of imaginary quadratic fields with the given class number and believes them to be complete. Infinitely many real quadratic fields with class number one Gauss conjectures that there are infinitely many real quadratic fields with class number one. The original Gauss class number problem for imaginary quadratic fields is significantly different and easier than the modern statement: he restricted to even discriminants, and allowed non-fundamental discriminants.
34.2 Status Gauss Conjecture Solved, Heilbronn, 1934. Low class number lists Class number 1: solved, Baker (1966), Stark (1967), Heegner (1952). Class number 2: solved, Baker (1971), Stark (1971)[2] Class number 3: solved, 1985[2] Class numbers h up to 100: solved, Watkins 2004[3] Infinitely many real quadratic fields with class number one Open. 125
126
CHAPTER 34. CLASS NUMBER PROBLEM
34.3 Lists of discriminants of class number 1 For more details on this topic, see Heegner number. For imaginary quadratic number fields, the (fundamental) discriminants of class number 1 are:
d = −3, −4, −7, −8, −11, −19, −43, −67, −163. The non-fundamental discriminants of class number 1 are:
d = −12, −16, −27, −28. Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss’s original question) are:
d = −4, −8, −12, −16, −28.
34.4 Modern developments In 1934, Hans Heilbronn proved the Gauss Conjecture. Equivalently, for any given class number, there are only finitely many imaginary quadratic number fields with that class number. Also in 1934, Heilbronn and Edward Linfoot showed that there were at most 10 imaginary quadratic number fields with class number 1 (the 9 known ones, and at most one further). The result was ineffective (see effective results in number theory): it did not allow bounds on the size of the remaining field. In later developments, the case n = 1 was first discussed by Kurt Heegner, using modular forms and modular equations to show that no further such field could exist. This work was not initially accepted; only with later work of Harold Stark and Bryan Birch was the position clarified, and Heegner’s work understood. See Stark–Heegner theorem, Heegner number. Practically simultaneously, Alan Baker proved what we now know as Baker’s theorem on linear forms in logarithms of algebraic numbers, which resolved the problem by a completely different method. The case n = 2 was tackled shortly afterwards, at least in principle, as an application of Baker’s work. (see Baker (1990).) √ The complete list of imaginary quadratic fields with class number one is Q( k) with k one of
−1, −2, −3, −7, −11, −19, −43, −67, −163. The general case awaited the discovery of Dorian Goldfeld that the class number problem could be connected to the L-functions of elliptic curves. This reduced the question, in principle, of effective determination, to one about establishing the existence of a multiple zero of such an L-function. This could be done on the basis of the later Gross-Zagier theorem. So at that point one could specify a finite calculation, the result of which would be a complete list for a given class number. In fact in practice such lists that are probably complete can be made by relatively simple methods; what is at issue is certainty. The cases up to n = 100 have now (2004) been done: see Watkins (2004).
34.5 Real quadratic fields The contrasting case of real quadratic fields is very different, and much less is known. That is because what enters the analytic formula for the class number is not h, the class number, on its own — but h log ε, where ε is a fundamental unit. This extra factor is hard to control. It may well be the case that class number 1 for real quadratic fields occurs infinitely often. The Cohen-Lenstra heuristics[4] are a set of more precise conjectures about the structure of class groups of quadratic fields. For real fields they predict that about 75.446% of the fields obtained by adjoining the square root of a prime will have class number 1, a result that agrees with computations.[5]
34.6. SEE ALSO
127
34.6 See also • List of number fields with class number one
34.7 Notes [1] The Gauss Class-Number Problems, by H. M. Stark [2] Ireland, K.; Rosen, M. (1993), A Classical Introduction to Modern Number Theory, New York, New York: Springer-Verlag, pp. 358–361, ISBN 0-387-97329-X [3] Watkins, M. (2004), Class numbers of imaginary quadratic fields, Mathematics of Computation 73, pp. 907–938 [4] Cohen, ch. 5.10 [5] te Riele & Williams
34.8 References • Goldfeld, Dorian (July 1985), “Gauss’ Class Number Problem For Imaginary Quadratic Fields” (PDF), Bulletin of the American Mathematical Society 13 (1): 23–37, doi:10.1090/S0273-0979-1985-15352-2 • Heegner, Kurt (1952), “Diophantische Analysis und Modulfunktionen”, Mathematische Zeitschrift 56 (3): 227– 253, doi:10.1007/BF01174749, MR 0053135 • te Riele, Herman; Williams, Hugh (2003), “New Computations Concerning the Cohen-Lenstra Heuristics” (PDF), Experimental Mathematics 12 (1): 99–113, doi:10.1080/10586458.2003.10504715 • Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Berlin: Springer, ISBN 3-54055640-0 • Baker, Alan (1990), Transcendental number theory, Cambridge Mathematical Library (2nd ed.), Cambridge University Press, ISBN 978-0-521-39791-9, MR 0422171
34.9 External links • Weisstein, Eric W., “Gauss’s Class Number Problem”, MathWorld.
Chapter 35
CM-field In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field. The abbreviation “CM” was introduced by (Shimura & Taniyama 1961).
35.1 Formal definition A number field K is a CM-field if it is a quadratic extension K/F where the base field F is totally real but K is totally imaginary. I.e., every embedding of F into C lies entirely within R , but there is no embedding of K into R . In other √ words, there is a subfield F of K such that K is generated over F by a single square root of an element, say β = α , in such a way that the minimal polynomial of β over the rational number field Q has all its roots non-real complex numbers. For this α should be chosen totally negative, so that for each embedding σ of K ′ into the real number field, σ(α) < 0.
35.2 Properties One feature of a CM-field is that complex conjugation on C induces an automorphism on the field which is independent of its embedding into C . In the notation given, it must change the sign of β. A number field K is a CM-field if and only if it has a “units defect”, i.e. if it contains a proper subfield F whose unit group has the same Z -rank as that of K (Remak 1954). In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet’s unit theorem.
35.3 Examples • The simplest, and motivating, example of a CM-field is an imaginary quadratic field, for which the totally real subfield is just the field of rationals • One of the most important examples of a CM-field is the cyclotomic field Q(ζn ) , which is generated by a primitive nth root of unity. It is a totally imaginary quadratic extension of the totally real field Q(ζn + ζn−1 ). The latter is the fixed field of complex conjugation, and Q(ζn ) is obtained from it by adjoining a square root of ζn2 + ζn−2 − 2 = (ζn − ζn−1 )2 . • The union QCM of all CM fields is similar to a CM field except that it has infinite degree. It is a quadratic extension of the union of all totally real fields QR . The absolute Galois group Gal(Q/QR ) is generated (as a closed subgroup) by all elements of order 2 in Gal(Q/Q), and Gal(Q/QCM ) is a subgroup of index 2. The Galois group Gal(QCM /Q) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(QR /Q). 128
35.4. REFERENCES
129
• If V is a complex abelian variety of dimension n, then any abelian algebra F of endomorphisms of V has rank at most 2n over Z. If it has rank 2n and V is simple then F is an order in a CM-field. Conversely any CM field arises like this from some simple complex abelian variety, unique up to isogeny.
35.4 References • Remak, Robert (1954), "Über algebraische Zahlkörper mit schwachem Einheitsdefekt”, Compositio Mathematica (in German) 12: 35–80, Zbl 0055.26805 • Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan 11, Princeton, N.J.: Princeton University Press • Shimura, Goro; Taniyama, Yutaka (1961), Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan 6, Tokyo: The Mathematical Society of Japan, MR 0125113 • Washington, Lawrence C. (1996). “Introduction to Cyclotomic fields” (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047.
Chapter 36
Compatible system of ℓ-adic representations In number theory, a compatible system of ℓ-adic representations is an abstraction of certain important families of ℓ-adic Galois representations, indexed by prime numbers ℓ, that have compatibility properties for almost all ℓ.
36.1 Examples Prototypical examples include the cyclotomic character and the Tate module of an abelian variety.
36.2 Variations A slightly more restrictive notion is that of a strictly compatible system of ℓ-adic representations which offers more control on the compatibility properties. More recently, some authors[1] have started requiring more compatibility related to p-adic Hodge theory.
36.3 Importance Compatible systems of ℓ-adic representations are a fundamental concept in contemporary algebraic number theory.
36.4 Notes [1] Such as Taylor 2004
36.5 References • Serre, Jean-Pierre (1998) [1968], Abelian l-adic representations and elliptic curves, Research Notes in Mathematics 7, with the collaboration of Willem Kuyk and John Labute, Wellesley, MA: A K Peters, ISBN 978-156881-077-5, MR 1484415 • Taylor, Richard (2004), “Galois representations”, Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6 13 (1): 73–119, MR 2060030
130
Chapter 37
Complete field In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers).
37.1 See also • Completion (ring theory) • Hensel’s lemma • Henselian ring • Ostrowski’s theorem
131
Chapter 38
Complex multiplication This article is about a topic in the theory of elliptic curves. For information about multiplication of complex numbers, see complex numbers. In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules). Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.[1]
38.1 Example of the imaginary quadratic field extension √ Consider an imaginary quadratic extension field K = Q( −d) d > 0 , which provides the typical example of complex multiplication. For the two periods ω1 , ω2 of an elliptic function f it is called to be of complex multiplication if there is an algebraic relation between f (z) and f (λz) for all λ in K . Conversely, Kronecker conjectured that every abelian extension of K would be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication, known as the Kronecker Jugendtraum and Hilbert’s twelfth problem. An example of an elliptic curve with complex multiplication is
C/Z[i] θ where Z[i] is the Gaussian integer ring, and θ is any non-zero complex number. Any such complex torus has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as
Y 2 = 4X 3 − aX having an order 4 automorphism sending
Y → −iY,
X → −X 132
38.1. EXAMPLE OF THE IMAGINARY QUADRATIC FIELD EXTENSION
133
An elliptic curve over the complex numbers is obtained as a quotient of the complex plane by a lattice Λ, here spanned by two fundamental periods ω1 and ω2 . The four-torsion is also shown, corresponding to the lattice 1/4 Λ containing Λ.
in line with the action of i on the Weierstrass elliptic functions. More generally, consider the group of lattice L on the complex plane generated by ω1 , ω2 . Then we define the Weierstrass function with a variable z in C as follows:
1 ℘(z; L) = ℘(z; ω1 , ω2 ) = 2 + z where
∑ n2 +m2 ̸=0
{
1 (z + mω1 + nω2 )2
−
}
1 (mω1 + nω2 )
2
134
CHAPTER 38. COMPLEX MULTIPLICATION
∑
g2 = 60
(mω1 + nω2 )−4
(m,n)̸=(0,0)
g3 = 140
∑
(mω1 + nω2 )−6 .
(m,n)̸=(0,0) ′
Let ℘ be the derivative of ℘ . Then we obtain the isomorphism:
z 7→ (1, ℘(z), ℘′ (z)) ∈ C3 which means the 1 to 1 correspondence between the complex torus group C/L and the elliptic curve { } E = x ∈ C3 x0 x22 = 4x31 − g2 x20 x1 − g3 x30 in the complex plane. This means that the ring of analytic automorphic group of C/L i.e., the ring of automorphisms of E , turn out to be isomorphic to the (subring of) integer rings of K . In particular, assume K ⊂ C and consider L as an ideal of K then C/L is agreed with the integer rings o . Rewrite τ = ω1 /ω2 Imτ > 0 and ∆(L) = g2 (L)3 − 27g3 (L)3 , then
J(τ ) = J(E) = J(L) = 26 33 g2 (L)3 /∆(L) . This means that the J-invariants of E belong to the algebraic numbers of K if E has complex multiplication.
38.2 Abstract theory of endomorphisms The ring of endomorphisms of an elliptic curve can be of one of three forms:the integers Z; an order in an imaginary quadratic number field; or an order in a definite quaternion algebra over Q.[2] When the field of definition is a finite field, there are always non-trivial endomorphisms of an elliptic curve, coming from the Frobenius map, so the complex multiplication case is in a sense typical (and the terminology isn't often applied). But when the base field is a number field, complex multiplication is the exception. It is known that, in a general sense, the case of complex multiplication is the hardest to resolve for the Hodge conjecture.
38.3 Kronecker and abelian extensions Kronecker first postulated that the values of elliptic functions at torsion points should be enough to generate all abelian extensions for imaginary quadratic fields, an idea that went back to Eisenstein in some cases, and even to Gauss. This became known as the Kronecker Jugendtraum; and was certainly what had prompted Hilbert’s remark above, since it makes explicit class field theory in the way the roots of unity do for abelian extensions of the rational number field, via Shimura’s reciprocity law. Indeed, let K be an imaginary quadratic field with class field H. Let E be an elliptic curve with complex multiplication by the integers of K, defined over H. Then the maximal abelian extension of K is generated by the x-coordinates of the points of finite order on some Weierstrass model for E over H.[3] Many generalisations have been sought of Kronecker’s ideas; they do however lie somewhat obliquely to the main thrust of the Langlands philosophy, and there is no definitive statement currently known.
38.4 Sample consequence It is no accident that
38.5. SINGULAR MODULI
eπ
√ 163
135
= 262537412640768743.99999999999925007 . . .
or equivalently,
eπ
√ 163
= 6403203 + 743.99999999999925007 . . .
is so close to an integer. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of modular forms, and the fact that [ Z
1+
√ ] −163 2
is a unique factorization domain. √ Here (1 + −163)/2 satisfies α² = α − 41. In general, S[α] denotes the set of all polynomial expressions in α with coefficients in S, which is the smallest ring containing α and S. Because α satisfies this quadratic equation, the required polynomials can be limited to degree one. Alternatively,
eπ
√ 163
= 123 (2312 − 1)3 + 743.99999999999925007 . . .
an internal structure due to certain Eisenstein series, and with similar simple expressions for the other Heegner numbers.
38.5 Singular moduli The points of the upper half-plane τ which correspond to the period ratios of elliptic curves over the complex numbers with complex multiplication are precisely the imaginary quadratic numbers.[4] The corresponding modular invariants j(τ) are the singular moduli, coming from an older terminology in which “singular” referred to the property of having non-trivial endomorphisms rather than referring to a singular curve.[5] The modular function j(τ) is algebraic on imaginary quadratic numbers τ:[6] these are the only algebraic numbers in the upper half-plane for which j is algebraic.[7] If Λ is a lattice with period ratio τ then we write j(Λ) for j(τ). If further Λ is an ideal a in the ring of integers OK of a quadratic imaginary field K then we write j(a) for the corresponding singular modulus. The values j(a) are then real algebraic integers, and generate the Hilbert class field H of K: the field extension degree [H:K] = h is the class number of K and the H/K is a Galois extension with Galois group isomorphic to the ideal class group of K. The class group acts on the values j(a) by [b] : j(a) → j(ab). In particular, if K has class number one, then j(a) = j(O) is a rational integer: for example, j(Z[i]) = j(i) = 1728.
38.6 See also • Abelian variety of CM-type, higher dimensions • Algebraic Hecke character • Heegner point • Hilbert’s twelfth problem • Lubin–Tate formal group, local fields • Drinfeld shtuka, global function field case
136
CHAPTER 38. COMPLEX MULTIPLICATION
38.7 Notes [1] Reid, Constance (1996), Hilbert, Springer, p. 200, ISBN 978-0-387-94674-0 [2] Silverman (1989) p.102 [3] Serre (1967) p.295 [4] Silverman (1986) p.339 [5] Silverman (1994) p.104 [6] Serre (1967) p.293 [7] Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 56. ISBN 0-521-20461-5. Zbl 0297.10013.
38.8 References • Borel, A.; Chowla, S.; Herz, C. S.; Iwasawa, K.; Serre, J.-P. Seminar on complex multiplication. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. Lecture Notes in Mathematics, No. 21 Springer-Verlag, Berlin-New York, 1966 • Husemöller, Dale H. (1987). Elliptic curves. Graduate Texts in Mathematics 111. With an appendix by Ruth Lawrence. Springer-Verlag. ISBN 0-387-96371-5. Zbl 0605.14032. • Lang, Serge (1983). Complex multiplication. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 255. New York: Springer-Verlag. ISBN 0-387-90786-6. Zbl 0536.14029. • Serre, J.-P. (1967). “XIII. Complex multiplication”. In Cassels, J.W.S.; Fröhlich, Albrecht. Algebraic Number Theory. Academic Press. pp. 292–296. • Shimura, Goro (1971). Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan 11. Tokyo: Iwanami Shoten. Zbl 0221.10029. • Shimura, Goro (1998). Abelian varieties with complex multiplication and modular functions. Princeton Mathematical Series 46. Princeton, NJ: Princeton University Press. ISBN 0-691-01656-9. Zbl 0908.11023. • Silverman, Joseph H. (1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106. SpringerVerlag. ISBN 0-387-96203-4. Zbl 0585.14026. • Silverman, Joseph H. (1994). Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 151. Springer-Verlag. ISBN 0-387-94328-5. Zbl 0911.14015.
38.9 External links • Complex multiplication from PlanetMath.org • Examples of elliptic curves with complex multiplication from PlanetMath.org • Ribet, Kenneth A. (October 1995). “Galois Representations and Modular Forms”. Bulletin of the American Mathematical Society 32 (4): 375–402. doi:10.1090/s0273-0979-1995-00616-6. CiteSeerX: 10.1.1.125.6114.
Chapter 39
Composite field (mathematics) A composite field is an object of study in field theory. Let L be a field, and let F, K be subfields of L. Then the (internal) composite of F and K is defined to be the intersection of all subfields of L containing both F and K. The composite is commonly denoted FK. When F and K are not regarded as subfields of a common field then the (external) composite is defined using the tensor product of fields.
39.1 References • Roman, Steven (1995). Field Theory. New York: Springer-Verlag. ISBN 0-387-94407-9., especially chapter 2
137
Chapter 40
Conductor (class field theory) In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.
40.1 Local conductor Let L/K be a finite abelian extension of non-archimedean local fields. The conductor of L/K, denoted f(L/K) , is the smallest non-negative integer n such that the higher unit group { } U (n) = 1 + mn = u ∈ O× : u ≡ 1 (mod mnK ) is contained in NL/K(L× ), where NL/K is field norm map and mK is the maximal ideal of K.[1] Equivalently, n is the (n) smallest integer such that the local Artin map is trivial on UK . Sometimes, the conductor is defined as mnK where n is as above.[2] The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero,[3] and it is tamely ramified if, and only if, the conductor is 1.[4] More precisely, the conductor computes the non-triviality of higher ramification groups: if s is the largest integer for which the "lower numbering" higher ramification group Gs is non-trivial, then f(L/K) = ηL/K (s) + 1 , where ηL/K is the function that translates from “lower numbering” to "upper numbering" of higher ramification groups.[5] The conductor of L/K is also related to the Artin conductors of characters of the Galois group Gal(L/K). Specifically,[6]
f(L/K)
mK
f
= lcm mKχ χ
where χ varies over all multiplicative complex characters of Gal(L/K), fχ is the Artin conductor of χ, and lcm is the least common multiple.
40.1.1
More general fields
The conductor can be defined in the same way for L/K a not necessarily abelian finite Galois extension of local fields.[7] However, it only depends on Lab /K, the maximal abelian extension of K in L, because of the “norm limitation theorem”, which states that, in this situation,[8][9] ( ) NL/K (L× ) = NLab /K (Lab )× . Additionally, the conductor can be defined when L and K are allowed to be slightly more general than local, namely if they are complete valued fields with quasi-finite residue field.[10] 138
40.2. GLOBAL CONDUCTOR
40.1.2
139
Archimedean fields
Mostly for the sake of global conductors, the conductor of the trivial extension R/R is defined to be 0, and the conductor of the extension C/R is defined to be 1.[11]
40.2 Global conductor 40.2.1
Algebraic number fields
The conductor of an abelian extension L/K of number fields can be defined, similarly to the local case, using the Artin map. Specifically, let θ : I m → Gal(L/K) be the global Artin map where the modulus m is a defining modulus for L/K; we say that Artin reciprocity holds for m if θ factors through the ray class group modulo m. We define the conductor of L/K, denoted f(L/K) , to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for f(L/K) , so it is the smallest such modulus.[12][13][14] Example • Taking as base the field of rational numbers, the Kronecker–Weber theorem states that an algebraic number field K is abelian over Q if and only if it is a subfield of a cyclotomic field Q(ζn ) .[15] The conductor of K is then the smallest such n. √ • Let L/K be Q( d)/Q where d is a squarefree integer. Then,[16]
) ∆ √ √ Q( d) f Q( d)/Q = ∞ ∆ √ Q( d) (
ford > 0 ford < 0
√ where ∆Q(√d) is the discriminant of Q( d)/Q . Relation to local conductors and ramification The global conductor is the product of local conductors:[17]
f(L/K) =
∏
pf(Lp /Kp ) .
p
As a consequence, a finite prime is ramified in L/K if, and only if, it divides f(L/K) .[18] An infinite prime v occurs in the conductor if, and only if, v is real and becomes complex in L.
40.3 Notes [1] Serre 1967, §4.2 [2] As in Neukirch 1999, definition V.1.6 [3] Neukirch 1999, proposition V.1.7 [4] Milne 2008, I.1.9 [5] Serre 1967, §4.2, proposition 1 [6] Artin & Tate 2009, corollary to theorem XI.14, p. 100 [7] As in Serre 1967, §4.2
140
CHAPTER 40. CONDUCTOR (CLASS FIELD THEORY)
[8] Serre 1967, §2.5, proposition 4 [9] Milne 2008, theorem III.3.5 [10] As in Artin & Tate 2009, §XI.4. This is the situation in which the formalism of local class field theory works. [11] Cohen 2000, definition 3.4.1 [12] Milne 2008, remark V.3.8 [13] Janusz 1973, pp. 158,168–169 [14] Some authors omit infinite places from the conductor, e.g. Neukirch 1999, §VI.6 [15] Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences 49 (Second ed.). pp. 155, 168. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002. [16] Milne 2008, example V.3.11 [17] For the finite part Neukirch 1999, proposition VI.6.5, and for the infinite part Cohen 2000, definition 3.4.1 [18] Neukirch 1999, corollary VI.6.6
40.4 References • Artin, Emil; Tate, John (2009) [1967], Class field theory, American Mathematical Society, ISBN 978-0-82184426-7, MR 2467155 • Cohen, Henri (2000), Advanced topics in computational number theory, Graduate Texts in Mathematics 193, Springer-Verlag, ISBN 978-0-387-98727-9 • Janusz, Gerald (1973), Algebraic Number Fields, Pure and Applied Mathematics 55, Academic Press, ISBN 0-12-380250-4, Zbl 0307.12001 • Milne, James (2008), Class field theory (v4.0 ed.), retrieved 2010-02-22 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Serre, Jean-Pierre (1967), “Local class field theory”, in Cassels, J. W. S.; Fröhlich, Albrecht, Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965, London: Academic Press, ISBN 0-12-163251-2, MR 0220701
Chapter 41
Conductor of an abelian variety In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how “bad” the bad reduction at some prime is. It is connected to the ramification in the field generated by the torsion points.
41.1 Definition For an abelian variety A defined over a field F as above, with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over Spec(R) (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism Spec(F) → Spec(R) gives back A. Let A0 denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a maximal ideal P of A with residue field k, A0 k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let uP be the dimension of the unipotent group and tP the dimension of the torus. The order of the conductor at P is
fP = 2uP + tP + δP , where δP ∈ N is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by
f=
∏
P fP .
P
41.2 Properties • A has good reduction at P if and only if uP = tP = 0 (which implies fP = δP = 0 ). • A has semistable reduction if and only if uP = 0 (then again δP = 0 ). • If A acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic at P, then δP = 0. • If p > 2d + 1, where d is the dimension of A, then δP = 0. 141
142
CHAPTER 41. CONDUCTOR OF AN ABELIAN VARIETY
41.3 References • S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 70–71. ISBN 3-540-61223-8. • J.-P. Serre; J. Tate (1968). “Good reduction of Abelian varieties”. Ann. Math. (The Annals of Mathematics, Vol. 88, No. 3) 88 (3): 492–517. doi:10.2307/1970722. JSTOR 1970722.
Chapter 42
Conductor-discriminant formula In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by Hasse (1926, 1930) for abelian extensions and by Artin (1931) for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension L/K of local or global fields from the Artin conductors of the irreducible characters Irr(G) of the Galois group G = G(L/K) .
42.1 Statement Let L/K be a finite Galois extension of global fields with Galois group G . Then the discriminant equals
∏
dL/K =
f(χ)χ(1) ,
χ∈Irr(G)
where f(χ) equals the global Artin conductor of χ .[1]
42.2 Example Let L = Q(ζpn )/Q be a cyclotomic extension of the rationals. The Galois group G equals (Z/pn )× . Because (p) is the only finite prime ramified, the global Artin conductor f(χ) equals the local one f(p) (χ) . Because G is abelian, every non-trivial irreducible character χ is of degree 1 = χ(1) . Then, the local Artin conductor of χ equals the conductor of the p -adic completion of Lχ = Lker(χ) /Q , i.e. (p)np , where np is the smallest natural (n ) number such that UQpp ⊆ NLχp /Qp (ULχp ) . If p > 2 , the Galois group G(Lp /Qp ) = G(L/Qp ) = (Z/pn )× is cyclic of order φ(pn ) , and by local class field theory and using that UQp /UQp = (Z/pk )× one sees easily that n f(p) (χ) = (pφ(p )(n−1/(p−1)) ) : the exponent is (k)
n−1 ∑
(φ(p ) − φ(p )) = nφ(p ) − 1 − (p − 1) n
i
i=0
n
n−2 ∑
pi = nφ(pn ) − pn−1 .
i=0
42.3 Notes [1] Neukirch 1999, VII.11.9.
143
144
CHAPTER 42. CONDUCTOR-DISCRIMINANT FORMULA
42.4 References • Artin, Emil (1931), “Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper.”, Journal für Reine und Angewandte Mathematik (in German) 164: 1–11, doi:10.1515/crll.1931.164.1, ISSN 00754102, Zbl 0001.00801 • Hasse, H. (1926), “Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie.”, Jahresbericht der Deutschen Mathematiker-Vereinigung (in German) 35: 1–55 • Hasse, H. (1930), “Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper.”, Journal für die reine und angewandte Mathematik (in German) 162: 169–184, doi:10.1515/crll.1930.162.169, ISSN 0075-4102 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
Chapter 43
Conjugate element (field theory) “Conjugate elements” redirects here. For conjugate group elements, see Conjugacy class. In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial pK,α(x) of α over K. Conjugate elements are also called Galois conjugates, or simply conjugates. Normally α itself is included in the set of conjugates of α.
43.1 Example The cube roots of the number one are: 1 √ 3 1 = − 12 + − 1 − 2
√ 3 2 i √ 3 2 i
The latter two roots are conjugate elements in L/K = Q[√3, i]/Q[√3] with minimal polynomial ( )2 1 3 x+ + = x2 + x + 1. 2 4
43.2 Properties If K is given inside an algebraically closed field C, then the conjugates can be taken inside C. If no such C is specified, one can take the conjugates in some relatively small field L. The smallest possible choice for L is to take a splitting field over K of pK,α, containing α. If L is any normal extension of K containing α, then by definition it already contains such a splitting field. Given then a normal extension L of K, with automorphism group Aut(L/K) = G, and containing α, any element g(α) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates. This follows as K(α) is K-isomorphic to K(β) by irreducibility of the minimal polynomial, and any isomorphism of fields F and F ' that maps polynomial p to p ' can be extended to an isomorphism of the splitting fields of p over F and p ' over F ', respectively. In summary, the conjugate elements of α are found, in any normal extension L of K that contains K(α), as the set of elements g(α) for g in Aut(L/K). The number of repeats in that list of each element is the separable degree [L:K(α)] ₑ . A theorem of Kronecker states that if α is a nonzero algebraic integer such that α and all of its conjugates in the complex numbers have absolute value at most 1, then α is a root of unity. There are quantitative forms of this, stating 145
146
CHAPTER 43. CONJUGATE ELEMENT (FIELD THEORY)
more precisely bounds (depending on degree) on the largest absolute value of a conjugate that imply that an algebraic integer is a root of unity.
43.3 References • David S. Dummit, Richard M. Foote, Abstract algebra, 3rd ed., Wiley, 2004.
43.4 External links • Weisstein, Eric W., “Conjugate Elements”, MathWorld.
Chapter 44
Cubic field In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.
44.1 Definition If K is a field extension of the rational numbers Q of degree [K:Q] = 3, then K is called a cubic field. Any such field is isomorphic to a field of the form
Q[x]/(f (x)) where f is an irreducible cubic polynomial with coefficients in Q. If f has three real roots, then K is called a totally real cubic field and it is an example of a totally real field. If, on the other hand, f has a non-real root, then K is called a complex cubic field. A cubic field K is called a cyclic cubic field, if it contains all three roots of its generating polynomial f. Equivalently, K is a cyclic cubic field if it is a Galois extension of Q, in which case its Galois group over Q is cyclic of order three. This can only happen if K is totally real. It is a rare occurrence in the sense that if the set of cubic fields is ordered by discriminant, then the proportion of cubic fields which are cyclic approaches zero as the bound on the discriminant approaches infinity.[1] √ A cubic field is called a pure cubic field, if it can be obtained by adjoining the real cube root 3 n of a cubefree positive integer n to the rational number field Q.
44.2 Examples √ • Adjoining the real cube root of 2 to the rational numbers gives the cubic field Q( 3 2) . This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant (in absolute value), namely −108.[2] • The complex cubic field obtained by adjoining to Q a root of x3 + x2 − 1 is not pure. It has the smallest discriminant (in absolute value) of all cubic fields, namely −23.[3] • Adjoining a root of x3 + x2 − 2x − 1 to Q yields a cyclic cubic field, and hence a totally real cubic field. It has the smallest discriminant of all totally real cubic fields, namely 49.[4] • The field obtained by adjoining to Q a root of x3 + x2 − 3x − 1 is an example of a totally real cubic field that is not cyclic. Its discriminant is 148, the smallest discriminant of a non-cyclic totally real cubic field.[5] • No cyclotomic fields are cubic because the degree of a cyclotomic field is equal to φ(n), where φ is Euler’s totient function, which only takes on even values (except for φ(1) = φ(2) = 1). 147
148
CHAPTER 44. CUBIC FIELD
44.3 Galois closure A cyclic cubic field K is its own Galois closure with Galois group Gal(K/Q) isomorphic to the cyclic group of order three. However, any other cubic field K is a non-galois extension of Q and has a field extension N of degree two as its Galois closure. The Galois group Gal(N/Q) is isomorphic to the symmetric group S 3 on three letters.
44.4 Associated quadratic field The discriminant of a cubic field K can be written uniquely as df 2 where d is a fundamental discriminant. Then, K is cyclic if, and only if, d = 1, in which case the only subfield of K is Q itself. If d ≠ 1, then the Galois closure N of K contains a unique quadratic field k whose discriminant is d (in the case d = 1, the subfield Q is sometimes considered as the “degenerate” quadratic field of discriminant 1). The conductor of N over k is f, and f 2 is the relative discriminant of N over k. The discriminant of N is d3 f 4 .[6][7] The field K is a pure cubic field if, and only if, d = −3. This is the case for which the quadratic field contained in the Galois closure of K is the cyclotomic field of cube roots of unity.[7]
44.5 Discriminant Since the sign of the discriminant of a number field K is (−1)r2 , where r2 is the number of conjugate pairs of complex embeddings of K into C, the discriminant of a cubic field will be positive precisely when the field is totally real, and negative if it is a complex cubic field. Given some real number N > 0 there are only finitely many cubic fields K whose discriminant DK satisfies |DK| ≤ N.[9] Formulae are known which calculate the prime decomposition of DK, and so it can be explicitly calculated.[10] However, it should be pointed out that, different from quadratic fields, several non-isomorphic cubic fields K 1 , ..., Km may share the same discriminant D. The number m of these fields is called the multiplicity[11] of the discriminant D. Some small examples are m = 2 for D = −1836, 3969, m = 3 for D = −1228, 22356, m = 4 for D = −3299, 32009, and m = 6 for D = −70956, 3054132. Any cubic field K will be of the form K = Q(θ) for some number θ that is a root of the irreducible polynomial
f (X) = X 3 − aX + b with a and b both being integers. The discriminant of f is Δ = 4a3 − 27b2 . Denoting the discriminant of K by D, the index i(θ) of θ is then defined by Δ = i(θ)2 D. In the case of a non-cyclic cubic field K this index formula can be combined with the conductor formula D = f 2 d to obtain a decomposition of the polynomial discriminant Δ = i(θ)2 f 2 d into the square of the product i(θ)f and the discriminant d of the quadratic field k associated with the cubic field K, where d is squarefree up to a possible factor 22 or 23 . Georgy Voronoy gave a method for separating i(θ) and f in the square part of Δ.[12] The study of the number of cubic fields whose discriminant is less than a given bound is a current area of research. Let N + (X) (respectively N − (X)) denote the number of totally real (respectively complex) cubic fields whose discriminant is bounded by X in absolute value. In the early 1970s, Harold Davenport and Hans Heilbronn determined the first term of the asymptotic behaviour of N ± (X) (i.e. as X goes to infinity).[13][14] By means of an analysis of the residue of the Shintani zeta function, combined with a study of the tables of cubic fields compiled by Karim Belabas (Belabas 1997) and some heuristics, David P. Roberts conjectured a more precise asymptotic formula:[15] 4ζ( 13 )B± 5 A± 6 X+ 5 X 2 3 12ζ(3) 5Γ( 3 ) ζ( 3 ) √ where A± = 1 or 3, B± = 1 or 3 , according to the totally real or complex case, ζ(s) is the Riemann zeta function, and Γ(s) is the Gamma function. A proof of this formula has been announced by Bhargava, Shankar & Tsimerman (2010) using methods based on Bhargava’s earlier work, as well as Taniguchi & Thorne (2011) based on the Shintani zeta function. N ± (X) ∼
44.6. UNIT GROUP
149
The blue crosses are the number of totally real cubic fields of bounded discriminant. The black line is the asymptotic distribution to first order whereas the green line includes the second order term.[8]
44.6 Unit group According to Peter Gustav Lejeune Dirichlet, the torsionfree unit rank r of an algebraic number field K with r1 real embeddings and r2 pairs of conjugate complex embeddings is determined by the formula r = r1 + r2 − 1. Hence a totally real cubic field K with r1 = 3, r2 = 0 has two independent units ε1 , ε2 and a complex cubic field K with r1 = r2 = 1 has a single fundamental unit ε1 . These fundamental systems of units can be calculated by means of generalized continued fraction algorithms by Voronoi,[16] which have been interpreted geometrically by Delone and Faddeev.[17]
44.7 Notes [1] Harvey Cohn computed an asymptotic for the number of cyclic cubic fields (Cohn 1954), while Harold Davenport and Hans Heilbronn computed the asymptotic for all cubic fields (Davenport & Heilbronn 1971). [2] Cohen 1993, §B.3 contains a table of complex cubic fields [3] Cohen 1993, §B.3 [4] Cohen 1993, §B.4 contains a table of totally real cubic fields and indicates which are cyclic [5] Cohen 1993, §B.4
150
CHAPTER 44. CUBIC FIELD
The blue crosses are the number of complex cubic fields of bounded discriminant. The black line is the asymptotic distribution to first order whereas the green line includes the second order term.[8]
[6] Hasse 1930 [7] Cohen 1993, §6.4.5 [8] The exact counts were computed by Michel Olivier and are available at . The first-order asymptotic is due to Harold Davenport and Hans Heilbronn (Davenport & Heilbronn 1971). The second-order term was conjectured by David P. Roberts (Roberts 2001) and a proof has been announced by Manjul Bhargava, Arul Shankar, and Jacob Tsimerman (Bhargava, Shankar & Tsimerman 2010). [9] H. Minkowski, Diophantische Approximationen, chapter 4, §5. [10] Llorente, P.; Nart, E. (1983). “Effective determination of the decomposition of the rational primes in a cubic field”. Proceedings of the American Mathematical Society 87 (4): 579–585. doi:10.1090/S0002-9939-1983-0687621-6. [11] Mayer, D. C. (1992). “Multiplicities of dihedral discriminants”. Math. Comp. 58 (198): 831–847 and S55–S58. doi:10.1090/S0025-5718-1992-1122071-3. [12] G. F. Voronoi, Concerning algebraic integers derivable from a root of an equation of the third degree, Master’s Thesis, St. Petersburg, 1894 (Russian). [13] Davenport & Heilbronn 1971 [14] Their work can also be interpreted as a computation of the average size of the 3-torsion part of the class group of a quadratic field, and thus constitutes one of the few proven cases of the Cohen–Lenstra conjectures: see, e.g. Bhargava,
44.8. REFERENCES
151
Manjul; Varma, Ila (2014), The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders, arXiv:1401.5875, This theorem [of Davenport and Heilbronn] yields the only two proven cases of the Cohen-Lenstra heuristics for class groups of quadratic fields. [15] Roberts 2001, Conjecture 3.1 [16] Voronoi, G. F. (1896). On a generalization of the algorithm of continued fractions (in Russian). Warsaw: Doctoral Dissertation. [17] Delone, B. N.; Faddeev, D. K. (1964). The theory of irrationalities of the third degree. Translations of Mathematical Monographs 10. Providence, Rhode Island: American Mathematical Society.
44.8 References • Şaban Alaca, Kenneth S. Williams, Introductory algebraic number theory, Cambridge University Press, 2004. • Belabas, Karim (1997), “A fast algorithm to compute cubic fields”, Mathematics of Computation 66 (219): 1213–1237, doi:10.1090/s0025-5718-97-00846-6, MR 1415795 • Bhargava, Manjul; Shankar, Arul; Tsimerman, Jacob (2010). “On the Davenport–Heilbronn theorem and second order terms”. arXiv:1005.0672. • Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Berlin, New York: Springer-Verlag, ISBN 978-3-540-55640-4, MR 1228206 • Cohn, Harvey (1954), “The density of abelian cubic fields”, Proceedings of the American Mathematical Society 5: 476–477, doi:10.2307/2031963, MR 0064076 • Davenport, Harold; Heilbronn, Hans (1971), “On the density of discriminants of cubic fields. II”, Proceedings of the Royal Society A 322 (1551): 405–420, doi:10.1098/rspa.1971.0075, MR 0491593 • Hasse, Helmut (1930), “Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage”, Mathematische Zeitschrift (in German) 31 (1): 565–582, doi:10.1007/BF01246435 • Roberts, David P. (2001), “Density of cubic field discriminants”, Mathematics of Computation 70 (236): 1699– 1705, doi:10.1090/s0025-5718-00-01291-6, MR 1836927 • Taniguchi, Takashi; Thorne, Frank (2011). “Secondary terms in counting functions for cubic fields”. arXiv:1102.2914.
Chapter 45
Cubic reciprocity Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word “reciprocity” comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is solvable.
45.1 History Sometime before 1748 Euler made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, after his death.[1] Gauss’s published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the Disquisitiones Arithmeticae (1801).[2] In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818)[3] he said that he was publishing these proofs because their techniques (Gauss’s lemma and Gaussian sums, respectively) can be applied to cubic and biquadratic reciprocity. Finally, a footnote in the second (of two) monographs on biquadratic reciprocity (1832) states that cubic reciprocity is most easily described in the ring of Eisenstein integers.[4] From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of cubic and biquadratic reciprocity around 1814.[5][6] Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein’s.[7] Jacobi published several theorems about cubic residuacity in 1827, but no proofs.[8] In his Königsberg lectures of 1836–37 Jacobi presented proofs.[7] The first published proofs were by Eisenstein (1844).[9][10][11]
45.2 Integers A cubic residue (mod p) is any number congruent to the third power of an integer (mod p). If x3 ≡ a (mod p) does not have an integer solution, a is a cubic nonresidue (mod p).[12] As is often the case in number theory, it is easiest to work modulo prime numbers, so in this section all moduli p, q, etc., are assumed to be positive, odd primes.[12] The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 2 (mod 3) every number is a cubic residue (mod q). Let q = 3n + 2; since 0 = 03 is obviously a cubic residue, assume x is not divisible by q. Then by Fermat’s little theorem,
xq = x3n+2 ≡ x (mod q) and xq−1 = x3n+1 ≡ 1 (mod q), so x = 1 · x ≡ xq xq−1 = x3n+2 x3n+1 = x6n+3 = (x2n+1 )3 is a cubic residue (mod q). 152
(mod q)
45.2. INTEGERS
153
Therefore, the only interesting case is when the modulus p ≡ 1 (mod 3). In this case, p ≡ 1 (mod 3), the nonzero residue classes (mod p) can be divided into three sets, each containing (p−1)/3 numbers. Let e be a cubic nonresidue. The first set is the cubic residues; the second one is e times the numbers in the first set, and the third is e2 times the numbers in the first set. Another way to describe this division is to let e be a primitive root (mod p); then the first (respectively second, third) set is the numbers whose indices with respect to this root are ≡ 0 (resp. 1, 2) (mod 3). In the vocabulary of group theory, the first set is a subgroup of index 3 (of the multiplicative group Z/pZ × ), and the other two are its cosets.
45.2.1
Primes ≡ 1 (mod 3)
A theorem of Fermat[13][14] states that every prime p ≡ 1 (mod 3) is the sum of a square and three times a square: p = a2 + 3b2 and (except for the signs of a and b) this representation is unique. Letting m = a + b and n = a − b, we see that this is equivalent to p = m2 − mn + n2 (which equals (n − m)2 − (n − m)n + n2 = m2 + m(n − m) + (n − m)2 , so m and n are not determined uniquely). Thus, 4p = (2m − n)2 + 3n2 = (2n − m)2 + 3m2 = (m + n)2 + 3(m − n)2 , and it is a straightforward exercise to show that exactly one of m, n, or m − n is a multiple of 3, so p=
1 4
( 2 ) L + 27M 2 , and this representation is unique up to the signs of L and M.[15]
For relatively prime integers m and n define the rational cubic residue symbol as {
[m] n
+1 if m is a cubic residue (mod n) −1 if m is a cubic nonresidue (mod n)
= 3
45.2.2
Euler
Euler’s conjectures[16][17] are based on the representation p = a2 + 3b2 . [ ] 2 p 3 [ ] 3 p 3 [ ] 5 p 3 [ ] 6 p 3
= 1 if and only if 3|b = 1 if and only if 9|b; or 9|(a ± b) [18]
= 1 if and only if 15|b; or 3|b and 5|a; or 15|(a ± b); or 15|(2a ± b) = 1 if and only if 9|b; or 9|(a ± 2b)
The first two can be restated as[19][20][21] • Let p ≡ 1 (mod 3) be a positive prime. Then 2 is a cubic residue of p if and only if p = a2 + 27b2 . • Let p ≡ 1 (mod 3) be a positive prime. Then 3 is a cubic residue of p if and only if 4p = a2 + 243b2 . Another conjecture of Euler is:[22] If
[ ] 7 p
= 1, then 3
(3|b and 7|a), or 21|(b ± a), or 7|(4b ± a), or 21|b, or 7|(b ± 2a).
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CHAPTER 45. CUBIC RECIPROCITY
45.2.3
Gauss
Gauss[23][24] proves that if p = 3n + 1 = [ ] M = 1 is an easy deduction. p
1 4
(
[ ] ) = L2 + 27M 2 , then L(n!)3 ≡ 1 (mod p), from which Lp 3
3
45.2.4
Jacobi
Jacobi stated (without proof)[25] Let q ≡ p ≡ 1 (mod 6) be positive primes, p =
1 4
( 2 ) L + 27M 2 , and let x be a solution of x2 ≡ −3 (mod q). Then
[ ] [ L+3M x ] [ ] L+3M x ( L−3M x ) p q 2 = 1 if and only if = 1 if and only if = 1. p 3 q q 3
3
(The “numerator” in the last expression is an integer (mod q), not a Legendre symbol). ( ) L′ If q = 41 L′2 + 27M ′2 , then x ≡ ± 3M ′ (mod q) , and we have [ LM ′ +L′ M ] [ ] ( LM ′ −L′ M ) q = 1 if and only if = 1. p 3 q 3
Along the same lines, von Lienen proved[26] ( ′ )2 ( ) ( ) +LM ′ ( L M2M ) p q = . q 3 p 3 p 3
( ) In this expression, the definition of pq is different from the one used elsewhere. To explain this, we split p and 3 q into p = ππ , q = ρρ , where π, π, ρ, ρ are Eisenstein primes ≡ 2 (mod 3)(that ) satisfy ρ ≡ N/2M (mod π), π ≡ N/2M′ (mod ρ), N = L′M + LM′, L ≡ L′ ≡ 1 (mod 3). Von Lienen defined pq as χᵨ(p), where χᵨ(·) denotes a (3 ) cubic residue character modulo ρ (see below). Note that the value of Li(p, q) = pq is not uniquely determined. 3 For example, when p = αα satisfies the above conditions, one can choose either α or α and call it π. However, once π is chosen, ρ ≡ N/2M (mod π) is uniquely determined, and both Li(x, q) and Li(x, p) are unambiguously defined for any x. Example: p = 13 = (−1+3ω)(−4−3ω), q = 79 = (−7+3ω)(−10−3ω), L = −5, M = 1, L′ = −17, M′ = 1. • One can choose π = −4−3ω and ρ = −7+3ω. Then χᵨ(p) = ω2 , χπ(q) = 1, χπ(N/2M) = ω, satisfying χᵨ(p) χπ(q) = (χπ(N/2M))2 , that is Li(p, q) Li(q, p) = (Li(N/2M, p))2 . • Alternatively, one can choose π = −1+3ω and ρ = −10−3ω. Then χᵨ(p) = ω, χπ(q) = 1, χπ(N/2M) = ω2 . These values are different from the previous ones, but they satisfy the same relationship.
45.2.5
Other theorems
Emma Lehmer proved[27]
q|LM or [ ] 9r ( ) L ≡ ± 2u+1 M (mod q), where Let q and p = 14 L2 + 27M 2 be primes. pq = 1 if and only if 3 u ≡ ̸ 0, 1, − 12 , − 31 (mod q) and 3u + 1 ≡ r2 (3u − 3) (mod q) Note that the first condition implies:
45.3. EISENSTEIN INTEGERS
155
Any number that divides L or M is a cubic residue (mod p). The first few examples[28] of this are equivalent to Euler’s conjectures: [ ] 2 p 3 [ ] 3 p [ ]3 5 p [ ]3 7 p 3
= 1 if and only if
L ≡ M ≡ 0 (mod 2)
= 1 if and only if
M ≡ 0 (mod 3)
= 1 if and only if
LM ≡ 0 (mod 5)
= 1 if and only if
LM ≡ 0 (mod 7)
Since obviously L ≡ M (mod 2), the criterion for q = 2 can be simplified as: [ ] 2 = 1 if and only if p 3
M ≡ 0 (mod 2)
Martinet proved[29] Let p ≡ q ≡ 1 (mod 3) be primes, pq =
1 4
(
) L2 + 27M 2 . Then
[ ] [ ] [ ] [ ] L q p L = 1 if and only if =1 p 3 q 3 p 3 q 3 Sharifi proved[30] Let p = 1 + 3x + 9x2 be prime. Then Any divisor of x is a cubic residue (mod p).
45.3 Eisenstein integers 45.3.1
Background
In his second monograph on biquadratic reciprocity, Gauss says: The theorems on biquadratic residues gleam with the greatest simplicity and genuine beauty only when the field of arithmetic is extended to imaginary numbers, so that without restriction, the numbers of the form a + bi constitute the object of study ... we call such numbers integral complex numbers.[31] [bold in the original] These numbers are now called the ring of Gaussian integers, denoted by Z[i]. Note that i is a fourth root of 1. In a footnote he adds The theory of cubic residues must be based in a similar way on a consideration of numbers of the form a + bh where h is an imaginary root of the equation h3 = 1 ... and similarly the theory of residues of higher powers leads to the introduction of other imaginary quantities.[32] In his first monograph on cubic reciprocity[33] Eisenstein developed the theory of the numbers built up from a cube root of unity; they are now called the ring of Eisenstein integers. Eisenstein said (paraphrasing) “to investigate the properties of this ring one need only consult Gauss’s work on Z[i] and modify the proofs”. This is not surprising since both rings are unique factorization domains. The “other imaginary quantities” needed for the “theory of residues of higher powers” are the rings of integers of the cyclotomic number fields; the Gaussian and Eisenstein integers are the simplest examples of these.
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CHAPTER 45. CUBIC RECIPROCITY
45.3.2
Facts and terminology √
3 Let ω = −1+i = e 3 be a complex cube root of unity. The Eisenstein integers Z[ω] are all numbers of the form 2 a + bω where a and b are ordinary integers. 2πi
Since ω3 − 1 = (ω − 1)(ω2 + ω + 1) = 0 and ω ≠ 1, we have ω2 = − ω − 1 and ω = − ω2 − 1.√Since ω 3 = ωω 2 = ωω = 1, ω = ω 2 and ω 2 = ω where the bar denotes complex conjugation. Also, ω − ω = i 3. If λ = a + bω and μ = c + dω, λ + μ = (a + c) + (b + d)ω and λ μ = ac + (ad + bc)ω + bdω2 = (ac − bd) + (ad + bc − bd)ω. This shows that Z[ω] is closed under addition and multiplication, making it a ring. The units are the numbers that divide 1. They are ±1, ±ω, and ±ω2 . They are similar to 1 and −1 in the ordinary integers, in that they divide every number. The units are the powers of −ω, a sixth (not just a third) root of unity. Given a number λ = a + bω, its conjugate means its complex conjugate a + bω2 = (a − b) − bω (not a − bω), and its associates are its six unit multiples:[34]
λ = a + bω ωλ = −b + (a − b)ω ω 2 λ = (b − a) − aω −λ = −a − bω −ωλ = b + (b − a)ω −ω 2 λ = (a − b) + aω The norm of λ = a + bω is the product of λ and its conjugate Nλ = λλ = a2 − ab + b2 . From the definition, if λ and μ are two Eisenstein integers, Nλμ = Nλ Nμ; in other words, the norm is a completely multiplicative function. The norm of zero is zero, the norm of any other number is a positive integer. ε is a unit if and only if Nε = 1. Note that the norm is always ≡ 0 or ≡ 1 (mod 3). Z[ω] is a unique factorization domain. The primes fall into three classes:[35] • 3 is a special case: 3 = −ω2 (1 − ω)2 . It is the only prime in Z divisible by the square of a prime in Z[ω]. In algebraic number theory, 3 is said to ramify in Z[ω]. • Positive primes in Z ≡ 2 (mod 3) are also primes in Z[ω]. In algebraic number theory, these primes are said to remain inert in Z[ω]. • Positive primes in Z ≡ 1 (mod 3) are the product of two conjugate primes in Z[ω]. p = Nπ = Nπ = ππ In algebraic number theory, these primes are said to split in Z[ω]. Thus, inert primes are 2, 5, 11, 17, ... and a factorization of the split primes is 7 = (3 + ω) × (2 − ω), 13 = (4 + ω) × (3 − ω), 19 = (3 − 2ω) × (5 + 2ω), 31 = (1 + 6ω) × (−5 − 6ω), ... The associates and conjugate of a prime are also primes. Note that the norm of an inert prime q is Nq = q2 ≡ 1 (mod 3). In order to state the unique factorization theorem, it is necessary to have a way of distinguishing one of the associates of a number. Eisenstein defines[36] a number to be primary if it is ≡ 2 (mod 3). It is straightforward to show that if
45.3. EISENSTEIN INTEGERS
157
gcd(Nλ, 3) = 1 then exactly one associate of λ is primary. A disadvantage of this definition is that the product of two primary numbers is the negative of a primary. Most modern authors[37][38] say that a number is primary if it is coprime to 3 and congruent to an ordinary integer (mod (1 − ω)2 ), which is the same as saying it is ≡ ±2 (mod 3). There are two reasons to do this: first, the product of two primaries is a primary, and second, it generalizes to all cyclotomic number fields.[37] Under this definition, if gcd(Nλ, 3) = 1 one of λ, ωλ, or ω2 λ is primary. A primary under Eisenstein’s definition is primary under the modern one, and if λ is primary under the modern one, either λ or −λ is primary under Eisenstein’s. Since −1 is a cube, this does not affect the statement of cubic reciprocity, but it does affect the unique factorization theorem. This article uses the modern definition, so The product of two primary numbers is primary and the conjugate of a primary number is also primary. The unique factorization theorem for Z[ω] is: if λ ≠ 0, then
λ = ±ω µ (1 − ω)ν π1α1 π2α2 π3α3 . . . where 0 ≤ μ ≤ 2, ν ≥ 0, each πi is a primary (under Eisenstein’s definition) prime, and each αi ≥ 1, and this representation is unique, up to the order of the factors. The notions of congruence[39] and greatest common divisor[40] are defined the same way in Z[ω] as they are for the ordinary integers Z. Because the units divide all numbers, a congruence (mod λ) is also true modulo any associate of λ, and any associate of a GCD is also a GCD.
45.3.3
Cubic residue character
An analogue of Fermat’s little theorem is true in Z[ω]: if α is not divisible by a prime π,[41]
αNπ−1 ≡ 1
(mod π)
Now assume that Nπ ≠ 3, so that Nπ ≡ 1 (mod 3). Then α
Nπ−1 3
makes sense, and α
Nπ−1 3
≡ ω k (mod π) for a unique unit ωk .
This unit is called the cubic residue character of α (mod π) and is denoted by[42] (α) π
3
= ωk ≡ α
Nπ−1 3
(mod π).
It has formal properties similar to those of the Legendre symbol.[43] The congruence x3 ≡ α (mod π) is solvable in Z[ω] if and only if (
αβ π
) 3
( ) ( ) α β = π π 3
( )
3
( )
α π
= 3
α π
where the bar denotes complex conjugation. 3
( ) if π and θ are associates,
( )
α π
= 3
( ) if α ≡ β (mod π),
α π 3
3
( ) =
α θ
β π 3
(α) π 3
= 1. [44]
158
CHAPTER 45. CUBIC RECIPROCITY
The cubic character can be extended multiplicatively to composite numbers (coprime to 3) in the “denominator” in the same way the Legendre symbol is generalized into the Jacobi symbol. Like the Jacobi symbol, if the “denominator” of the cubic character is composite, then if the “numerator” is a cubic residue mod the “denominator” the symbol will equal 1, if the symbol does not equal 1 then the “numerator” is a cubic nonresidue, but the symbol can equal 1 when the “numerator” is a nonresidue: (
(α) λ 3
=
α π1
)α 1 ( 3
α π2
)α 2 3
. . . where λ = π1α1 π2α2 π3α3 . . .
If a and b are ordinary integers, gcd(a, b) = gcd(b, 3) = 1, then[45][46]
45.3.4
(a) b 3
= 1.
Statement of the theorem
Let α and β be primary. Then ( ) ( ) α β = . β α 3
3
There are supplementary theorems[47][48] for the units and the prime 1 − ω: Let α = a + bω be primary, a = 3m + 1 and b = 3n. (If a ≡ 2 (mod 3) replace α with its associate −α; this will not change the value of the cubic characters.) Then ( ) ( ) ( ) 1−a−b a−1 b ω 1−ω 3 −m−n m , =ω 3 =ω =ω 3 =ω , = ω 3 = ωn . α α α 3
3
45.4 See also • Quadratic reciprocity • Quartic reciprocity • Eisenstein reciprocity • Artin reciprocity
45.5 Notes [1] Euler, Tractatus ..., §§ 407–410 [2] Gauss, DA, footnote to art. 358 [3] Gauss, Theorematis fundamentalis ... [4] Gauss, BQ, § 30 [5] Cox, pp. 83–90 [6] Lemmermeyer, pp. 199–201, 222–224 [7] Lemmermeyer, p. 200 [8] Jacobi, De residuis cubicis .... [9] Eisenstein, Beweis des Reciprocitätssatzes ... [10] Eisenstein, Nachtrag zum cubischen...
3
45.5. NOTES
[11] Eisenstein, Application de l'algèbre... [12] cf. Gauss, BQ § 2 [13] Gauss, DA, Art. 182 [14] Cox, Ex. 1.4–1.5 [15] Ireland & Rosen, Props 8.3.1 & 8.3.2 [16] Euler, Tractatus, §§ 407–401 [17] Lemmermeyer, p. 222–223 [18] The symbol m|n is read "m divides n" and means there is an k such that n = m·k. [19] Cox, p. 2, Thm. 4.15, Ex. 4.15 [20] Ireland & Rosen, Prop. 9.6.2, Ex 9.23 [21] Lemmermeyer, Prop. 7.1 & 7.2 [22] Tractatus de numerorum doctrina capita sedecim, quae supersunt, 411, footnote (chapter 11) [23] Gauss, DA footnote to art. 358 [24] Lemmermeyer, Ex. 7.9 [25] Jacobi, De residuis cubicis... [26] Lemmermeyer, p. 226–227 [27] Lemmermeyer, Prop.7.4 [28] Lemmermeyer, pp. 209–212, Props 7.1-7.3 [29] Lemmermeyer, Ex. 7.11 [30] Lemmermeyer, Ex. 7.12 [31] Gauss, BQ, § 30, translation in Cox, p. 83 [32] Gauss, BQ, § 30, translation in Cox, p. 84 [33] Ireland & Rosen p. 14 [34] Ireland & Rosen Prop. 9.3.5 [35] Ireland & Rosen Prop 9.1.4 [36] Ireland & Rosen, p. 113 [37] Ireland & Rosen, p. 206 [38] Lemmermeyer, p. 361 calls such numbers semi-primary. [39] cf. Gauss, BQ, §§ 38–45 [40] cf. Gauss, BQ, §§ 46–47 [41] Ireland & Rosen. Prop 9.3.1 [42] Ireland & Rosen, p. 112 [43] Ireland & Rosen, Prop 9.3.3 [44] Ireland & Rosen, Prop. 9.3.3 [45] Ireland & Rosen, Prop. 9.3.4 [46] Lemmermeyer, Prop 7.7 [47] Lemmermeyer, Th. 6.9 [48] Ireland & Rosen, Ex. 9.32–9.37
159
160
CHAPTER 45. CUBIC RECIPROCITY
45.6 References The references to the original papers of Euler, Jacobi, and Eisenstein were copied from the bibliographies in Lemmermeyer and Cox, and were not used in the preparation of this article.
45.6.1
Euler
• Euler, Leonhard (1849), Tractatus de numeroroum doctrina capita sedecim quae supersunt, Comment. Arithmet. 2 This was actually written 1748–1750, but was only published posthumously; It is in Vol V, pp. 182–283 of • Euler, Leonhard (1911–1944), Opera Omnia, Series prima, Vols I–V, Leipzig & Berlin: Teubner
45.6.2
Gauss
The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form “Gauss, BQ, § n". Footnotes referencing the Disquisitiones Arithmeticae are of the form “Gauss, DA, Art. n". • Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6 • Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7 These are in Gauss’s Werke, Vol II, pp. 65–92 and 93–148 Gauss’s fifth and sixth proofs of quadratic reciprocity are in • Gauss, Carl Friedrich (1818), Theoramatis fundamentalis in doctrina de residuis quadraticis demonstrationes et amplicationes novae This is in Gauss’s Werke, Vol II, pp. 47–64 German translations of all three of the above are the following, which also has the Disquisitiones Arithmeticae and Gauss’s other papers on number theory. • Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8
45.6.3
Eisenstein
• Eisenstein, Ferdinand Gotthold (1844), Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus den dritten Wurzeln der Einheit zusammengesetzen Zahlen, J. Reine Angew. Math. 27, pp. 289–310 (Crelle’s Journal) • Eisenstein, Ferdinand Gotthold (1844), Nachtrag zum cubischen Reciprocitätssatzes für die aus den dritten Wurzeln der Einheit zusammengesetzen Zahlen, Criterien des cubischen Characters der Zahl 3 and ihrer Teiler, J. Reine Angew. Math. 28, pp. 28–35 (Crelle’s Journal) • Eisenstein, Ferdinand Gotthold (1845), Application de l'algèbre à l'arithmétique transcendante, J. Reine Angew. Math. 29 pp. 177–184 (Crelle’s Journal) These papers are all in Vol I of his Werke.
45.7. EXTERNAL LINKS
45.6.4
161
Jacobi
• Jacobi, Carl Gustave Jacob (1827), De residuis cubicis commentatio numerosa, J. Reine Angew. Math. 2 pp. 66–69 (Crelle’s Journal) This is in Vol VI of his Werke
45.6.5
Modern authors
• Cox, David A. (1989), Primes of the form x2 + n y2 , New York: Wiley, ISBN 0-471-50654-0 • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X • Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-54066957-4
45.7 External links • Weisstein, Eric W., “Cubic Reciprocity Theorem”, MathWorld.
Chapter 46
Cyclotomic character In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space is generally denoted by R(1) (that is, it is a representation χ : G → AutR(R(1)) ≈ GL(1, R)).
46.1 p-adic cyclotomic character If p is a prime, and G is the absolute Galois group of the rational numbers, the p-adic cyclotomic character is a group homomorphism
χp : G → Z× p where Zp× is the group of units of the ring of p-adic integers. This homomorphism is defined as follows. Let ζn be a primitive pn root of unity. Every pn root of unity is a power of ζn uniquely defined as an element of the ring of integers modulo pn . Primitive roots of unity correspond to the invertible elements, i.e. to (Z/pn )× . An element g of the Galois group G sends ζn to another primitive pn root of unity
θ = ζnag,n where ag,n ∈ (Z/pn )× . For a given g, as n varies, the ag,n form a comptatible system in the sense that they give an element of the inverse limit of the (Z/pn )× , which is Z × . Therefore, the p-adic cyclotomic character sends g to the system (ag,n)n, thus encoding the action of g on all p-power roots of unity. In fact, χp is a continuous homomorphism (where the topology on G is the Krull topology, and that on Zp× is the p-adic topology).
46.2 As a compatible system of ℓ-adic representations By varying ℓ over all prime numbers, a compatible system of ℓ-adic representations is obtained from the ℓ-adic cyclotomic characters (when considering compatible systems of representations, the standard terminology is to use the symbol ℓ to denote a prime instead of p). That is to say, χ = { χℓ }ℓ is a “family” of ℓ-adic representations
χℓ : GQ → GL1 (Zℓ ) satisfying certain compatibilities between different primes. In fact, the χℓ form a strictly compatible system of ℓ-adic representations. 162
46.3. GEOMETRIC REALIZATIONS
163
46.3 Geometric realizations The p-adic cyclotomic character is the p-adic Tate module of the multiplicative group scheme Gm,Q over Q. As such, its representation space can be viewed as the inverse limit of the groups of pn th roots of unity in Q. In terms of cohomology, the p-adic cyclotomic character is the dual of the first p-adic étale cohomology group of Gm. It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of H 2 é ( P1 ). In terms of motives, the p-adic cyclotomic character is the p-adic realization of the Tate motive Z(1). As a Grothendieck motive, the Tate motive is the dual of H 2 ( P1 ).[1]
46.4 Properties The p-adic cyclotomic character satisfies several nice properties. • It is unramified at all primes ℓ ≠ p (i.e. the inertia subgroup at ℓ acts trivially). • If Frobℓ is a Frobenius element for ℓ ≠ p, then χ (Frobℓ) = ℓ • It is crystalline at p.
46.5 See also • Tate twist
46.6 References [1] Section 3 of Deligne, Pierre (1979), “Valeurs de fonctions L et périodes d'intégrales”, in Borel, Armand; Casselman, William, Automorphic Forms, Representations, and L-Functions, Proceedings of the Symposium in Pure Mathematics (in French) 33.2, Providence, RI: AMS, p. 325, ISBN 0-8218-1437-0, MR 0546622, Zbl 0449.10022
Chapter 47
Cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers. The n-th cyclotomic field Q(ζn) (where n > 2) is obtained by adjoining a primitive n-th root of unity ζn to the rational numbers. The cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat’s last theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n) – and more precisely, because of the failure of unique factorization in their rings of integers – that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences.
47.1 Properties A cyclotomic field is the splitting field of the cyclotomic polynomial
Φn (x) =
∏
(
k
x − e2iπ n
)
1≤k≤n
gcd(k,n)=1
and therefore it is a Galois extension of the field of rational numbers. The degree of the extension [Q(ζn):Q] is given by φ(n) where φ is Euler’s phi function. A complete set of Galois conjugates is given by { (ζn)a } , where a runs over the set of invertible residues modulo n (so that a is relative prime to n). The Galois group is naturally isomorphic to the multiplicative group (Z/nZ)× of invertible residues modulo n, and it acts on the primitive nth roots of unity by the formula b: (ζn)a → (ζn)a b .
47.2 Relation with regular polygons Gauss made early inroads in the theory of cyclotomic fields, in connection with the geometrical problem of constructing a regular n-gon with a compass and straightedge. His surprising result that had escaped his predecessors was that a regular heptadecagon (with 17 sides) could be so constructed. More generally, if p is a prime number, then a regular p-gon can be constructed if and only if p is a Fermat prime; in other words if φ(p) = 2k is a power of 2. For n = 3 and n = 6 primitive roots of unity admit a simple expression via square root of three, namely: 164
47.3. RELATION WITH FERMAT’S LAST THEOREM
165
ζ3 = √3 i − 1/2, ζ6 = √3 i + 1/2 Hence, both corresponding cyclotomic fields are identical to the quadratic field Q(√−3). In the case of ζ4 = i = √−1 the identity of Q(ζ4 ) to a quadratic field is even more obvious. This is not the case for n = 5 though, because expressing roots of unity requires square roots of quadratic integers, that means that roots belong to a second iteration of quadratic extension. The geometric problem for a general n can be reduced to the following question in Galois theory: can the nth cyclotomic field be built as a sequence of quadratic extensions?
47.3 Relation with Fermat’s Last Theorem A natural approach to proving Fermat’s Last Theorem is to factor the binomial xn + yn , where n is an odd prime, appearing in one side of Fermat’s equation xn + yn = zn as follows: xn + yn = (x + y) (x + ζy) … (x + ζn − 1 y). Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q(ζn). If unique factorization of algebraic integers were true, then it could have been used to rule out the existence of nontrivial solutions to Fermat’s equation. Several attempts to tackle Fermat’s Last Theorem proceeded along these lines, and both Fermat’s proof for n = 4 and Euler’s proof for n = 3 can be recast in these terms. Unfortunately, the unique factorization fails in general – for example, for n = 23 – but Kummer found a way around this difficulty. He introduced a replacement for the prime numbers in the cyclotomic field Q(ζp), expressed the failure of unique factorization quantitatively via the class number hp and proved that if hp is not divisible by p (such numbers p are called regular primes) then Fermat’s theorem is true for the exponent n = p. Furthermore, he gave a criterion to determine which primes are regular and using it, established Fermat’s theorem for all prime exponents p less than 100, with the exception of the irregular primes 37, 59, and 67. Kummer’s work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by Iwasawa in Iwasawa theory and by Kubota and Leopoldt in their theory of p-adic zeta functions.
47.3.1
List of Class Numbers to Cyclotomic Field
(sequence A061653 in OEIS), or
A055513 (for prime n)
47.4 See also • Kronecker–Weber theorem • Cyclotomic polynomial
47.5 References • Bryan Birch, “Cyclotomic fields and Kummer extensions”, in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.III, pp. 45–93. • Daniel A. Marcus, Number Fields, third edition, Springer-Verlag, 1977 • Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83 (2 ed.), New York: Springer-Verlag, ISBN 0-387-94762-0, MR 1421575 • Serge Lang, Cyclotomic Fields I and II, Combined second edition. With an appendix by Karl Rubin. Graduate Texts in Mathematics, 121. Springer-Verlag, New York, 1990. ISBN 0-387-96671-4
166
CHAPTER 47. CYCLOTOMIC FIELD
47.6 Further reading • Coates, John; Sujatha, R. (2006). Cyclotomic Fields and Zeta Values. Springer Monographs in Mathematics. Springer-Verlag. ISBN 3-540-33068-2. Zbl 1100.11002. • Weisstein, Eric W., “Cyclotomic Field”, MathWorld. • Hazewinkel, Michiel, ed. (2001), “Cyclotomic field”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4
Chapter 48
Cyclotomic unit In mathematics, a cyclotomic unit (or circular unit) is a unit of an algebraic number field which is the product of numbers of the form (ζa n − 1) for ζ n an nth root of unity and 0 < a < n. Note that if n is the power of a prime ζa n − 1 itself is not a unit; however the numbers (ζa n − 1)/(ζ n − 1) for (a, n) = 1, and ±ζa n generate the group of cyclotomic units in this case (n power of a prime). The cyclotomic units form a subgroup of finite index in the group of units of a cyclotomic field. The index of this subgroup of real cyclotomic units (those cyclotomic units in the maximal real subfield) within the full unit group is equal to the class number of the maximal real subfield of the cyclotomic field.[1] Note also that if n is a composite number, the subgroup of cyclotomic units generated by (ζa n − 1)/(ζ n − 1)with (a, n) = 1 is not of finite index in general.[2] The cyclotomic units satisfy ∏ distribution relations. Let a be a rational number prime to p and let ga denote exp(2πia)−1. Then for a≠ 0 we have pb=a gb = ga .[3] Using these distribution relations and the symmetry relation ζa n − 1 = -ζa n (ζ-a n − 1) a basis Bn of the cyclotomic units can be constructed with the property that Bd ⊆ Bn for d | n.[4]
48.1 See also • Elliptic unit • Modular unit
48.2 References [1] Washington, Theorem 8.2 [2] Washington, 8.8, page 150, for n equal to 55. [3] Lang (1990) p.157 [4] http://perisic.com/cyclotomic
• Lang, Serge (1990). Cyclotomic Fields I and II. Graduate Texts in Mathematics 121 (second combined ed.). Springer Verlag. ISBN 3-540-96671-4. Zbl 0704.11038. 167
168
CHAPTER 48. CYCLOTOMIC UNIT
• Narkiewicz, Władysław (1990). Elementary and analytic theory of numbers (Second, substantially revised and extended ed.). Springer-Verlag. ISBN 3-540-51250-0. Zbl 0717.11045. • Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Graduate Texts in Mathematics 83 (2nd ed.). Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047.
Chapter 49
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.
49.1 The prehistory of Dedekind domains In the 19th century it became a common technique to gain insight into integral solutions of polynomial equations using rings of algebraic numbers of higher degree. For instance, fix a positive integer m . In the attempt to determine which√integers are represented by the quadratic form x2 + my 2 , it is natural to factor the quadratic form √ √ into (x + −my)(x − −my) , the factorization taking place in the ring of integers of the quadratic field Q( −m) . Similarly, for a positive integer n the polynomial z n − y n (which is relevant for solving the Fermat equation xn + y n = z n ) can be factored over the ring Z[ζn ] , where ζn is a primitive n root of unity. For a few small values of m and n these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of Fermat ( m = 1, n = 4 ) and Euler ( m = 2, 3, n = √ 3 ). By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field Q( D) is a PID was well known to the quadratic form theorists. Especially, Gauss had looked at the case of imaginary quadratic fields: he found exactly nine values of D < 0 for which the ring of integers is a PID and conjectured that there are no further values. (Gauss’ conjecture was proven more than one hundred years later by Kurt Heegner, Alan Baker and Harold Stark.) However, this was understood (only) in the language of equivalence classes of quadratic forms, so that in particular the analogy between quadratic forms and the Fermat equation seems not to have been perceived. In 1847 Gabriel Lamé announced a solution of Fermat’s Last Theorem for all n > 2 —i.e., that the Fermat equation has no solutions in nonzero integers—but it turned out that his solution hinged on the assumption that the cyclotomic ring Z[ζn ] is a UFD. Ernst Kummer had shown three years before that this was not the case already for n = 23 (the full, finite list of values for which Z[ζn ] is a UFD is now known). At the same time, Kummer developed powerful new methods to prove Fermat’s Last Theorem at least for a large class of prime exponents n using what we now recognize as the fact that the ring Z[ζn ] is a Dedekind domain. In fact Kummer worked not with ideals but with “ideal numbers”, and the modern definition of an ideal was given by Dedekind. By the 20th century, algebraists and number theorists had come to realize that the condition of being a PID is rather delicate, whereas the condition of being a Dedekind domain is quite robust. For instance the ring of ordinary integers is a PID, but as seen above the ring OK of algebraic integers in a number field K need not be a PID. In fact, although √ Gauss also conjectured that there are infinitely many primes p such that the ring of integers of Q( p) is a PID, to this day we do not even know whether there are infinitely many number fields K (of arbitrary degree) such that OK 169
170
CHAPTER 49. DEDEKIND DOMAIN
is a PID! On the other hand, the ring of integers in a number field is always a Dedekind domain. Another illustration of the delicate/robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property -- a Noetherian domain R is Dedekind iff for every maximal ideal M of R the localization RM is a Dedekind ring. But a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring (DVR), so the same local characterization cannot hold for PIDs: rather, one may say that the concept of a Dedekind ring is the globalization of that of a DVR.
49.2 Alternative definitions For an integral domain R that is not a field, all of the following conditions are equivalent: (DD1) Every nonzero proper ideal factors into primes. (DD2) R is Noetherian, and the localization at each maximal ideal is a Discrete Valuation Ring. (DD3) Every nonzero fractional ideal of R is invertible. (DD4) R is an integrally closed, Noetherian domain with Krull dimension one (i.e., every nonzero prime ideal is maximal). Thus a Dedekind domain is a domain that satisfies any one, and hence all four, of (DD1) through (DD4). Which of these conditions one takes as the definition is therefore merely a matter of taste. In practice, it is often easiest to verify (DD4). A Krull domain is a higher-dimensional analog of a Dedekind domain: a Dedekind domain that is not a field is a Krull domain of dimension 1. This notion can be used to study the various characterizations of a Dedekind domain. In fact, this is the definition of a Dedekind domain used in Bourbaki’s “Commutative algebra”. A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; i.e., every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective.[1]
49.3 Some examples of Dedekind domains All principal ideal domains and therefore all discrete valuation rings are Dedekind domains. The ring R = OK of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one (to see the last property, observe that for any nonzero prime ideal I of R, R/I is finite and recall that a finite integral domain is a field), so by (DD4) R is a Dedekind domain. As above, this includes all the examples considered by Kummer and Dedekind and was the motivating case for the general definition, and these remain among the most studied examples. The other class of Dedekind rings that is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic curve over a field k. Then the coordinate ring k[C] of regular functions on C is a Dedekind domain. Indeed, this is essentially an algebraic translation of these geometric terms: the coordinate ring of any affine variety is, by definition, a finitely generated k-algebra, so Noetherian; moreover curve means dimension one and nonsingular implies (and, in dimension one, is equivalent to) normal, which by definition means integrally closed. Both of these constructions can be viewed as special cases of the following basic result: Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.[2] Applying this theorem when R is itself a PID gives us a way of building Dedekind domains out of PIDs. Taking R = Z this construction tells us precisely that rings of integers of number fields are Dedekind domains. Taking R = k[t] gives us the above case of nonsingular affine curves. Zariski and Samuel were sufficiently taken by this construction to pose as a question whether every Dedekind domain arises in such a fashion, i.e., by starting with a PID and taking the integral closure in a finite degree field extension.[3] A surprisingly simple negative answer was given by L. Claborn.[4] If the situation is as above but the extension L of K is algebraic of infinite degree, then it is still possible for the
49.4. FRACTIONAL IDEALS AND THE CLASS GROUP
171
integral closure S of R in L to be a Dedekind domain, but it is not guaranteed. For example, take again R = Z, K = Q and now take L to be the field Q of all algebraic numbers. The integral closure is nothing else than the ring Z of all algebraic integers. Since the square root of an algebraic integer is again an algebraic integer, it is not possible to factor any nonzero nonunit algebraic integer into a finite product of irreducible elements, which implies that Z is not Noetherian! In general, the integral closure of a Dedekind domain in an infinite algebraic extension is a Prüfer domain; it turns out that the ring of algebraic integers is slightly more special than this: it is a Bézout domain.
49.4 Fractional ideals and the class group Let R be an integral domain with fraction field K. A fractional ideal is a nonzero R-submodule I of K for which there exists a nonzero x in K such that xI ⊂ R. ∑ Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums n in jn , in ∈ I, jn ∈ J : the product IJ is again a fractional ideal. The set Frac(R) of all fractional ideals endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal R. For any fractional ideal I, one may define the fractional ideal
I ∗ = (R : I) = {x ∈ K | xI ⊂ R}. One then tautologically has I ∗ I ⊂ R . In fact one has equality if and only if I, as an element of the monoid of Frac(R), is invertible. In other words, if I has any inverse, then the inverse must be I ∗ . A principal fractional ideal is one of the form xR for some nonzero x in K. Note that each principal fractional ideal is invertible, the inverse of xR being simply x1 R . We denote the subgroup of principal fractional ideals by Prin(R). A domain R is a PID if and only if every fractional ideal is principal. In this case, we have Frac(R) = Prin(R) = K × /R× , since two principal fractional ideals xR and yR are equal iff xy −1 is a unit in R. For a general domain R, it is meaningful to take the quotient of the monoid Frac(R) of all fractional ideals by the submonoid Prin(R) of principal fractional ideals. However this quotient itself is generally only a monoid. In fact it is easy to see that the class of a fractional ideal I in Frac(R)/Prin(R) is invertible if and only if I itself is invertible. Now we can appreciate (DD3): in a Dedekind domain—and only in a Dedekind domain! -- is every fractional ideal invertible. Thus these are precisely the class of domains for which Frac(R)/Prin(R) forms a group, the ideal class group Cl(R) of R. This group is trivial if and only if R is a PID, so can be viewed as quantifying the obstruction to a general Dedekind domain being a PID. We note that for an arbitrary domain one may define the Picard group Pic(R) as the group of invertible fractional ideals Inv(R) modulo the subgroup of principal fractional ideals. For a Dedekind domain this is of course the same as the ideal class group. However, on a more general class of domains—including Noetherian domains and Krull domains -- the ideal class group is constructed in a different way, and there is a canonical homomorphism
→ which is however generally neither injective nor surjective. This is an affine analogue of the distinction between Cartier divisors and Weil divisors on a singular algebraic variety. A remarkable theorem of L. Claborn (Claborn 1966) asserts that for any abelian group G whatsoever, there exists a Dedekind domain R whose ideal class group is isomorphic to G. Later, C.R. Leedham-Green showed that such an R may constructed as the integral closure of a PID in a quadratic field extension (Leedham-Green 1972). In 1976, M. Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain that is a subring of the rational function field of an elliptic curve, and conjectured that such an “elliptic” construction should be possible for a general abelian group (Rosen 1976). Rosen’s conjecture was proven in 2008 by P.L. Clark (Clark 2009). In contrast, one of the basic theorems in algebraic number theory asserts that the class group of the ring of integers of a number field is finite; its cardinality is called the class number and it is an important and rather mysterious invariant, notwithstanding the hard work of many leading mathematicians from Gauss to the present day.
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CHAPTER 49. DEDEKIND DOMAIN
49.5 Finitely generated modules over a Dedekind domain In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain. Let us briefly recall the structure theory in the case of a finitely generated module M over a PID R . We define the torsion submodule T to be the set of elements m of M such that rm = 0 for some nonzero r in R . Then: (M1) T can be decomposed into a direct sum of cyclic torsion modules, each of the form R/I for some nonzero ideal I of R . By the Chinese Remainder Theorem, each R/I can further be decomposed into a direct sum of submodules of the form R/P i , where P i is a power of a prime ideal. This decomposition need not be unique, but any two decompositions T ∼ = R/P1a1 ⊕ · · · ⊕ R/Prar ∼ = R/Qb11 ⊕ · · · ⊕ R/Qbss differ only in the order of the factors. (M2) The torsion submodule is a direct summand: i.e., there exists a complementary submodule P of M such that M =T ⊕P . (M3PID) P isomorphic to Rn for a uniquely determined non-negative integer n . In particular, P is a finitely generated free module. Now let M be a finitely generated module over an arbitrary Dedekind domain R . Then (M1) and (M2) hold verbatim. However, it follows from (M3PID) that a finitely generated torsionfree module P over a PID is free. In particular, it asserts that all fractional ideals are principal, a statement that is false whenever R is not a PID. In other words, the nontriviality of the class group Cl(R) causes (M3PID) to fail. Remarkably, the additional structure in torsionfree finitely generated modules over an arbitrary Dedekind domain is precisely controlled by the class group, as we now explain. Over an arbitrary Dedekind domain one has (M3DD) P is isomorphic to a direct sum of rank one projective modules: P ∼ = I1 ⊕ · · · ⊕ Ir . Moreover, for any rank one projective modules I1 , . . . , Ir , J1 , . . . , Js , one has I1 ⊕ · · · ⊕ Ir ∼ = J1 ⊕ · · · ⊕ Js if and only if
r=s and I1 ⊗ · · · ⊗ Ir ∼ = J1 ⊗ · · · ⊗ Js . Rank one projective modules can be identified with fractional ideals, and the last condition can be rephrased as
[I1 · · · Ir ] = [J1 · · · Js ] ∈ Cl(R). Thus a finitely generated torsionfree module of rank n > 0 can be expressed as Rn−1 ⊕ I , where I is a rank one projective module. The Steinitz class for P over R is the class [I] of I in Cl(R): it is uniquely determined.[5] A consequence of this is: Theorem: Let R be a Dedekind domain. Then K0 (R) ∼ = Z ⊕ Cl(R) , where K0 (R) is the Grothendieck group of the commutative monoid of finitely generated projective R modules. These results were established by Ernst Steinitz in 1912. An additional consequence of this structure, which is not implicit in the preceding theorem, is that if the two projective modules over a Dedekind domain have the same class in the Grothendieck group, then they are in fact abstractly isomorphic.
49.6. LOCALLY DEDEKIND RINGS
173
49.6 Locally Dedekind rings There exist integral domains R that are locally but not globally Dedekind: the localization of R at each maximal ideal is a Dedekind ring (equivalently, a DVR) but R itself is not Dedekind. As mentioned above, such a ring cannot be Noetherian. It seems that the first examples of such rings were constructed by N. Nakano in 1953. In the literature such rings are sometimes called “proper almost Dedekind rings.”
49.7 Notes [1] Cohn 2003, 2.4. Exercise 9 [2] The theorem follows, for instance, from the Krull–Akizuki theorem. [3] Zariski and Samuel, p. 284 [4] Claborn 1965, Example 1-9 [5] Fröhlich & Taylor (1991) p.95
49.8 References • Bourbaki, Nicolas (1972), Commutative Algebra, Addison-Wesley • Claborn, Luther (1965), “Dedekind domains and rings of quotients”, Pacific J. Math. 15: 59–64, doi:10.2140/pjm.1965.15.59 • Claborn, Luther (1966), “Every abelian group is a class group”, Pacific J. Math. 18: 219–222, doi:10.2140/pjm.1966.18.219 • Clark, Pete L. (2009), “Elliptic Dedekind domains revisited” (PDF), L'Enseignement Mathematique 55: 213– 225, doi:10.4171/lem/55-3-1 • Cohn, Paul M. (2003). Further algebra and applications. Springer. ISBN 1-85233-667-6. • Fröhlich, A.; Taylor, M.J. (1991), “II. Dedekind domains”, Algebraic number theory, Cambridge studies in advanced mathematics 27, Cambridge University Press, pp. 35–101, ISBN 0-521-36664-X, Zbl 0744.11001 • Leedham-Green, C.R. (1972), “The class group of Dedekind domains”, Trans. Amer. Math. Soc. 163: 493– 500, doi:10.2307/1995734, JSTOR 1995734 • Nakano, Noburu (1953), “Idealtheorie in einem speziellen unendlichen algebraischen Zahlkörper”, J. Sci. Hiroshima Univ. Ser. A. 16: 425–439 • Rosen, Michael (1976), “Elliptic curves and Dedekind domains”, Proc. Amer. Math. Soc. 57 (2): 197–201, doi:10.2307/2041187, JSTOR 2041187 • Steinitz, E. (1912), “Rechteckige Systeme und Moduln in algebraischen Zahlkörpern”, Math. Ann. 71 (3): 328–354, doi:10.1007/BF01456849 • Zariski, Oscar; Samuel, Pierre (1958), Commutative Algebra, Volume I, D. Van Nostrand Company
49.9 Further reading • Edwards, Harold M. (1990), Divisor theory, Boston: Birkhäuser Verlag, ISBN 0-8176-3448-7, Zbl 0689.12001
49.10 External links • Hazewinkel, Michiel, ed. (2001), “Dedekind ring”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4
Chapter 50
Dedekind zeta function In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q. In particular, it can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2. The Dedekind zeta function is named for Richard Dedekind who introduced them in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie.[1]
50.1 Definition and basic properties Let K be an algebraic number field. Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series
ζK (s) =
∑ I⊆OK
1 (NK/Q (I))s
where I ranges through the non-zero ideals of the ring of integers OK of K and NK/Q(I) denotes the absolute norm of I (which is equal to both the index [OK : I] of I in OK or equivalently the cardinality of quotient ring OK / I). This sum converges absolutely for all complex numbers s with real part Re(s) > 1. In the case K = Q, this definition reduces to that of the Riemann zeta function.
50.1.1
Euler product
The Dedekind zeta function of K has an Euler product which is a product over all the prime ideals P of OK
ζK (s) =
∏ P ⊆OK
1 1 − (NK/Q (P ))−s
, Re for (s) > 1.
This is the expression in analytic terms of the uniqueness of prime factorization of the ideals I in OK. The fact that, for Re(s) > 1, ζK(s) is given by a product of non-zero numbers implies that it is non-zero in this region.
50.1.2
Analytic continuation and functional equation
Erich Hecke first proved that ζK(s) has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at s = 1. The residue at that pole is given by the analytic class number formula and is made up of important arithmetic data involving invariants of the unit group and class group of K. 174
50.2. SPECIAL VALUES
175
The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let ΔK denote discriminant of K, let r1 (resp. r2 ) denote the number of real places (resp. complex places) of K, and let ΓR (s) = π −s/2 Γ(s/2) and ΓC (s) = 2(2π)−s Γ(s) where Γ(s) is the Gamma function. Then, the function ΛK (s) = |∆K |
s/2
ΓR (s)r1 ΓC (s)r2 ζK (s)
satisfies the functional equation ΛK (s) = ΛK (1 − s).
50.2 Special values Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h(K) of K, the regulator R(K) of K, the number w(K) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of OK and the leading term is given by
lim s−r ζK (s) = −
s→0
h(K)R(K) . w(K)
Combining the functional equation and the fact that Γ(s) is infinite at all integers less than or equal to zero yields that ζK(s) vanishes at all negative even integers. It even vanishes at all negative odd integers unless K is totally real (i.e. r2 = 0; e.g. Q or a real quadratic field). In the totally real case, Carl Ludwig Siegel showed that ζK(s) is a non-zero rational number at negative odd integers. Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the algebraic K-theory of K.
50.3 Relations to other L-functions For the case in which K is an abelian extension of Q, its Dedekind zeta function can be written as a product of Dirichlet L-functions. For example, when K is a quadratic field this shows that the ratio ζK (s) ζQ (s) is the L-function L(s, χ), where χ is a Jacobi symbol used as Dirichlet character. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet L-function is an analytic formulation of the quadratic reciprocity law of Gauss. In general, if K is a Galois extension of Q with Galois group G, its Dedekind zeta function is the Artin L-function of the regular representation of G and hence has a factorization in terms of Artin L-functions of irreducible Artin representations of G. L (s) The relation with Artin L-functions shows that if L/K is a Galois extension then ζζK (s) is holomorphic ( ζK (s) “divides” ζL (s) ): for general extensions the result would follow from the Artin conjecture for L-functions.[2]
Additionally, ζK(s) is the Hasse–Weil zeta function of Spec OK [3] and the motivic L-function of the motive coming from the cohomology of Spec K.[4]
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50.4 Arithmetically equivalent fields Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. Wieb Bosma and Bart de Smit (2002) used Gassmann triples to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.
50.5 Notes [1] Narkiewicz 2004, §7.4.1 [2] Martinet (1977) p.19 [3] Deninger 1994, §1 [4] Flach 2004, §1.1
50.6 References • Bosma, Wieb; de Smit, Bart (2002), “On arithmetically equivalent number fields of small degree”, in Kohel, David R.; Fieker, Claus, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci. 2369, Berlin, New York: Springer-Verlag, pp. 67–79, doi:10.1007/3-540-45455-1_6, ISBN 978-3-540-43863-2, MR 2041074 • Section 10.5.1 of Cohen, Henri (2007), Number theory, Volume II: Analytic and modern tools, Graduate Texts in Mathematics 240, New York: Springer, doi:10.1007/978-0-387-49894-2, ISBN 978-0-387-49893-5, MR 2312338 • Deninger, Christopher (1994), "L-functions of mixed motives”, in Jannsen, Uwe; Kleiman, Steven; Serre, JeanPierre, Motives, Part 1, Proceedings of Symposia in Pure Mathematics 55.1, American Mathematical Society, pp. 517–525, ISBN 978-0-8218-1635-6 • Flach, Mathias, “The equivariant Tamagawa number conjecture: a survey”, in Burns, David; Popescu, Christian; Sands, Jonathan et al., Stark’s conjectures: recent work and new directions (PDF), Contemporary Mathematics 358, American Mathematical Society, pp. 79–125, ISBN 978-0-8218-3480-0 • Martinet, J. (1977), “Character theory and Artin L-functions”, in Fröhlich, A., Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975, Academic Press, pp. 1–87, ISBN 0-12-268960-7, Zbl 0359.12015 • Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag, Chapter 7, ISBN 978-3-540-21902-6, MR 2078267
Chapter 51
Degree of a field extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the “size” of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.
51.1 Definition and notation Suppose that E/F is a field extension. Then E may be considered as a vector space over F (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by [E:F]. The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension E/F is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements). The degree should not be confused with the transcendence degree of a field; for example, the field Q(X) of rational functions has infinite degree over Q, but transcendence degree only equal to 1.
51.2 The multiplicativity formula for degrees Given three fields arranged in a tower, say K a subfield of L which is in turn a subfield of M, there is a simple relation between the degrees of the three extensions L/K, M/L and M/K:
[M : K] = [M : L] · [L : K]. In other words, the degree going from the “bottom” to the “top” field is just the product of the degrees going from the “bottom” to the “middle” and then from the “middle” to the “top”. It is quite analogous to Lagrange’s theorem in group theory, which relates the order of a group to the order and index of a subgroup — indeed Galois theory shows that this analogy is more than just a coincidence. The formula holds for both finite and infinite degree extensions. In the infinite case, the product is interpreted in the sense of products of cardinal numbers. In particular, this means that if M/K is finite, then both M/L and L/K are finite. If M/K is finite, then the formula imposes strong restrictions on the kinds of fields that can occur between M and K, via simple arithmetical considerations. For example, if the degree [M:K] is a prime number p, then for any intermediate field L, one of two things can happen: either [M:L] = p and [L:K] = 1, in which case L is equal to K, or [M:L] = 1 and [L:K] = p, in which case L is equal to M. Therefore there are no intermediate fields (apart from M and K themselves). 177
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CHAPTER 51. DEGREE OF A FIELD EXTENSION
51.2.1
Proof of the multiplicativity formula in the finite case
Suppose that K, L and M form a tower of fields as in the degree formula above, and that both d = [L:K] and e = [M:L] are finite. This means that we may select a basis {u1 , ..., ud} for L over K, and a basis {w1 , ..., we} for M over L. We will show that the elements umwn, for m ranging through 1, 2, ..., d and n ranging through 1, 2, ..., e, form a basis for M/K; since there are precisely de of them, this proves that the dimension of M/K is de, which is the desired result. First we check that they span M/K. If x is any element of M, then since the wn form a basis for M over L, we can find elements an in L such that
x=
e ∑
an wn = a1 w1 + · · · + ae we .
n=1
Then, since the um form a basis for L over K, we can find elements bm,n in K such that for each n,
an =
d ∑
bm,n um = b1,n u1 + · · · + bd,n ud .
m=1
Then using the distributive law and associativity of multiplication in M we have
x=
(
e ∑ n=1
d ∑
) bm,n um
wn =
m=1
e ∑ d ∑
bm,n (um wn ),
n=1 m=1
which shows that x is a linear combination of the umwn with coefficients from K; in other words they span M over K. Secondly we must check that they are linearly independent over K. So assume that
0=
e ∑ d ∑
bm,n (um wn )
n=1 m=1
for some coefficients bm,n in K. Using distributivity and associativity again, we can group the terms as
0=
e ∑
(
n=1
d ∑
) bm,n um
wn ,
m=1
and we see that the terms in parentheses must be zero, because they are elements of L, and the wn are linearly independent over L. That is,
0=
d ∑
bm,n um
m=1
for each n. Then, since the bm,n coefficients are in K, and the um are linearly independent over K, we must have that bm,n = 0 for all m and all n. This shows that the elements umwn are linearly independent over K. This concludes the proof.
51.2.2
Proof of the formula in the infinite case
In this case, we start with bases uα and wᵦ of L/K and M/L respectively, where α is taken from an indexing set A, and β from an indexing set B. Using an entirely similar argument as the one above, we find that the products uαwᵦ form a basis for M/K. These are indexed by the cartesian product A × B, which by definition has cardinality equal to the product of the cardinalities of A and B.
51.3. EXAMPLES
179
51.3 Examples • The complex numbers are a field extension over the real numbers with degree [C:R] = 2, and thus there are no non-trivial fields between them. • The field extension Q(√2, √3), obtained by adjoining √2 and √3 to the field Q of rational numbers, has degree 4, that is, [Q(√2, √3):Q] = 4. The intermediate field Q(√2) has degree 2 over Q; we conclude from the multiplicativity formula that [Q(√2, √3):Q(√2)] = 4/2 = 2. • The finite field GF(125) = GF(53 ) has degree 3 over its subfield GF(5). More generally, if p is a prime and n, m are positive integers with n dividing m, then [GF(pm ):GF(pn )] = m/n. • The field extension C(T)/C, where C(T) is the field of rational functions over C, has infinite degree (indeed it is a purely transcendental extension). This can be seen by observing that the elements 1, T, T 2 , etc., are linearly independent over C. • The field extension C(T 2 ) also has infinite degree over C. However, if we view C(T 2 ) as a subfield of C(T), then in fact [C(T):C(T 2 )] = 2. More generally, if X and Y are algebraic curves over a field K, and F : X → Y is a surjective morphism between them of degree d, then the function fields K(X) and K(Y) are both of infinite degree over K, but the degree [K(X):K(Y)] turns out to be equal to d.
51.4 Generalization Given two division rings E and F with F contained in E and the multiplication and addition of F being the restriction of the operations in E, we can consider E as a vector space over F in two ways: having the scalars act on the left, giving a dimension [E:F] , and having them act on the right, giving a dimension [E:F]ᵣ. The two dimensions need not agree. Both dimensions however satisfy a multiplication formula for towers of division rings; the proof above applies to left-acting scalars without change.
51.5 References • page 215, Jacobson, N. (1985). Basic Algebra I. W. H. Freeman and Company. ISBN 0-7167-1480-9. Proof of the multiplicativity formula. • page 465, Jacobson, N. (1989). Basic Algebra II. W. H. Freeman and Company. ISBN 0-7167-1933-9. Briefly discusses the infinite dimensional case.
Chapter 52
Different ideal In algebraic number theory, the different ideal (sometimes simply the different) is defined to measure the (possible) lack of duality in the ring of integers of an algebraic number field K, with respect to the field trace. It then encodes the ramification data for prime ideals of the ring of integers. It was introduced by Richard Dedekind in 1882.[1][2]
52.1 Definition If OK is the ring of integers of K, and tr denotes the field trace from K to the rational number field Q, then
x 7→ tr x2 is an integral quadratic form on OK. Its discriminant as quadratic form need not be +1 (in fact this happens only for the case K = Q). Define the inverse different or codifferent [3][4] or Dedekind’s complementary module[5] as the set I of x ∈ K such that tr(xy) is an integer for all y in OK, then I is a fractional ideal of K containing OK. By definition, the different ideal δK is the inverse fractional ideal I −1 : it is an ideal of OK. The ideal norm of δK is equal to the ideal of Z generated by the field discriminant DK of K. The different of an element α of K with minimal polynomial f is defined to be δ(α) = f′(α) if α generates the field K (and zero otherwise):[6] we may write
δ(α) =
∏(
α − α(i)
)
where the α(i) run over all the roots of the characteristic polynomial of α other than α itself.[7] The different ideal is generated by the differents of all integers α in OK.[6][8] This is Dedekind’s original definition.[9] The different is also defined for an finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields.
52.2 Relative different The relative different δL / K is defined in a similar manner for an extension of number fields L / K. The relative norm of the relative different is then equal to the relative discriminant ΔL / K.[10] In a tower of fields L / K / F the relative differents are related by δL / F = δL / KδK / F.[5][11] The relative different equals the annihilator of the relative Kähler differential module Ω1OL /OK :[10][12] δL/K = {x ∈ OL : xdy = 0 all for y ∈ OL }. The ideal class of the relative different δL / K is always a square in the class group of OL, the ring of integers of L.[13] Since the relative discriminant is the norm of the relative different it is the square of a class in the class group of OK:[14] indeed, it is the square of the Steinitz class for OL as a OK-module.[15] 180
52.3. RAMIFICATION
181
52.3 Ramification The relative different encodes the ramification data of the field extension L / K. A prime ideal p of K ramifies in L if the factorisation of p in L contains a prime of L to a power higher than 1: this occurs if and only if p divides the relative discriminant ΔL / K. More precisely, if p = P 1 e(1) ... Pke(k) is the factorisation of p into prime ideals of L then Pi divides the relative different δL / K if and only if Pi is ramified, that is, if and only if the ramification index e(i) is greater than 1.[11][16] The precise exponent to which a ramified prime P divides δ is termed the differential exponent of P and is equal to e − 1 if P is tamely ramified: that is, when P does not divide e.[17] In the case when P is wildly ramified the differential exponent lies in the range e to e + νP(e) − 1.[16][18][19] The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions:[20]
∞ ∑ (|Gi | − 1). i=0
52.4 Local computation The different may be defined for an extension of local fields L / K. In this case we may take the extension to be simple, generated by a primitive element α which also generates a power integral basis. If f is the minimal polynomial for α then the different is generated by f'(α).
52.5 Notes [1] Dedekind 1882 [2] Bourbaki 1994, p. 102 [3] Serre 1979, p. 50 [4] Fröhlich & Taylor 1991, p. 125 [5] Neukirch 1999, p. 195 [6] Narkiewicz 1990, p. 160 [7] Hecke 1981, p. 116 [8] Hecke 1981, p. 121 [9] Neukirch 1999, pp. 197–198 [10] Neukirch 1999, p. 201 [11] Fröhlich & Taylor 1991, p. 126 [12] Serre 1979, p. 59 [13] Hecke 1981, pp. 234–236 [14] Narkiewicz 1990, p. 304 [15] Narkiewicz 1990, p. 401 [16] Neukirch 1999, pp. 199 [17] Narkiewicz 1990, p. 166 [18] Weiss 1976, p. 114 [19] Narkiewicz 1990, pp. 194,270 [20] Weiss 1976, p. 115
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52.6 References • Bourbaki, Nicolas (1994), Elements of the history of mathematics, Berlin: Springer-Verlag, ISBN 978-3-54064767-6, MR 1290116. Translated from the original French by John Meldrum • Dedekind, Richard (1882), "Über die Discriminanten endlicher Körper”, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 29 (2): 1–56. Retrieved 5 August 2009 • Fröhlich, Albrecht; Taylor, Martin (1991), Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, ISBN 0-521-36664-X, Zbl 0744.11001 • Hecke, Erich (1981), Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics 77, Translated by George U. Brauer and Jay R. Goldman with the assistance of R. Kotzen, New York–Heidelberg–Berlin: Springer-Verlag, ISBN 3-540-90595-2, Zbl 0504.12001 • Narkiewicz, Władysław (1990), Elementary and analytic theory of algebraic numbers (2nd, substantially revised and extended ed.), Springer-Verlag; PWN-Polish Scientific Publishers, ISBN 3-540-51250-0, Zbl 0717.11045 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Translated from the French by Marvin Jay Greenberg, Springer-Verlag, ISBN 0-387-90424-7, Zbl 0423.12016 • Weiss, Edwin (1976), Algebraic Number Theory (2nd unaltered ed.), Chelsea Publishing, ISBN 0-8284-02930, Zbl 0348.12101
Chapter 53
Differential Galois theory In mathematics, differential Galois theory studies the Galois groups of differential equations.
53.1 Overview Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory. The problem of finding which integrals of elementary functions can be expressed with other elementary functions is analogous to the problem of solutions of polynomial equations by radicals in algebraic Galois theory, and is solved by Picard–Vessiot theory.
53.2 Definitions For any differential field F, there is a subfield Con(F) = {f in F | Df = 0}, called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is a simple transcendental extension of F (i.e. G = F(t) for some transcendental t) such that Dt = Ds/s for some s in F. This has the form of a logarithmic derivative. Intuitively, one may think of t as the logarithm of some element s of F, in which case, this condition is analogous to the ordinary chain rule. But it must be remembered that F is not necessarily equipped with a unique logarithm; one might adjoin many “logarithm-like” extensions to F. Similarly, an exponential extension is a simple transcendental extension which satisfies Dt = tDs. With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, G is called a Liouvillian differential extension of F if there is a finite chain of subfields from F to G where each extension in the chain is either algebraic, logarithmic, or exponential.
53.3 See also • Liouville’s theorem (differential algebra) • Risch algorithm 183
184
CHAPTER 53. DIFFERENTIAL GALOIS THEORY
53.4 References • Bertrand, D. (1996), “Review of “Lectures on differential Galois theory"" (PDF), Bulletin of the American Mathematical Society 33 (2), doi:10.1090/s0273-0979-96-00652-0, ISSN 0002-9904 • Beukers, Frits (1992), “8. Differential Galois theory”, in Waldschmidt, Michel; Moussa, Pierre; Luck, JeanMarc; Itzykson, Claude, From number theory to physics. Lectures of a meeting on number theory and physics held at the Centre de Physique, Les Houches (France), March 7–16, 1989, Berlin: Springer-Verlag, pp. 413– 439, ISBN 3-540-53342-7, Zbl 0813.12001 • Magid, Andy R. (1994), Lectures on differential Galois theory, University Lecture Series 7, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-7004-4, MR 1301076 • Magid, Andy R. (1999), “Differential Galois theory” (PDF), Notices of the American Mathematical Society 46 (9): 1041–1049, ISSN 0002-9920, MR 1710665 • van der Put, Marius; Singer, Michael F. (2003), Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 328, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44228-8, MR 1960772
Chapter 54
Discrete valuation In mathematics, a discrete valuation is an integer valuation on a field K, that is a function ν : K → Z ∪ {∞} satisfying the conditions ν(x · y) = ν(x) + ν(y) { } ν(x + y) ≥ min ν(x), ν(y) ν(x) = ∞ ⇐⇒ x = 0 for all x, y ∈ K . Note that often the trivial valuation which takes on only the values 0, ∞ is explicitly excluded. A field with a non-trivial discrete valuation is called a discrete valuation field.
54.1 Discrete valuation rings and valuations on fields To every field with discrete valuation ν we can associate the subring
OK := {x ∈ K | ν(x) ≥ 0} of K , which is a discrete valuation ring. Conversely, the valuation ν : A → Z ∪ {∞} on a discrete valuation ring A can be extended in a unique way to a discrete valuation on the quotient field K = Quot(A) ; the associated discrete valuation ring OK is just A .
54.2 Examples • For a fixed prime p and for any element x ∈ Q different from zero write x = pj ab with j, a, b ∈ Z such that p does not divide a, b , then ν(x) = j is a discrete valuation on Q , called the p-adic valuation. • Given a Riemann surface X , we can consider the field K = M (X) of meromorphic functions X → C ∪ {∞} . For a fixed point p ∈ X , we define a discrete valuation on K as follows: ν(f ) = j if and only if j is the largest integer such that the function f (z)/(z − p)j can be extended to a holomorphic function at p . This means: if ν(f ) = j > 0 then f has a root of order j at the point p ; if ν(f ) = j < 0 then f has a pole of order −j at p . In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point p on the curve. More examples can be found in the article on discrete valuation rings. 185
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54.3 References • Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3259-2, MR 1915966
Chapter 55
Discriminant In algebra, the discriminant of a polynomial is a function of its coefficients, typically denoted by a capital 'D' or the capital Greek letter Delta (Δ). It gives information about the nature of its roots. Typically, the discriminant is zero if and only if the polynomial has a multiple root. For example, the discriminant of the quadratic polynomial
ax2 + bx + c is
∆ = b2 − 4ac. Here for real a, b and c, if Δ > 0, the polynomial has two real roots, if Δ = 0, the polynomial has one real double root, and if Δ < 0, the two roots of the polynomial are complex conjugates. The discriminant of the cubic polynomial
ax3 + bx2 + cx + d is
∆ = b2 c2 − 4ac3 − 4b3 d − 27a2 d2 + 18abcd. For higher degrees, the discriminant is always a polynomial function of the coefficients. It becomes significantly longer for the higher degrees. The discriminant of a general quartic has 16 terms,[1] that of a quintic has 59 terms,[2] that of a 6th degree polynomial has 246 terms,[3] and the number of terms increases exponentially with the degree. A polynomial has a multiple root (i.e. a root with multiplicity greater than one) in the complex numbers if and only if its discriminant is zero. The concept also applies if the polynomial has coefficients in a field which is not contained in the complex numbers. In this case, the discriminant vanishes if and only if the polynomial has a multiple root in any algebraically closed field containing the coefficients. As the discriminant is a polynomial function of the coefficients, it is defined as long as the coefficients belong to an integral domain R and, in this case, the discriminant is in R. In particular, the discriminant of a polynomial with integer coefficients is always an integer. This property is widely used in number theory. The term “discriminant” was coined in 1851 by the British mathematician James Joseph Sylvester.[4]
55.1 Definition In terms of the roots, the discriminant is given by 187
188
∆ = a2n−2 n
CHAPTER 55. DISCRIMINANT
∏
(ri − rj )2 = (−1)n(n−1)/2 a2n−2 n
i
∏
(ri − rj )
i̸=j
where an is the leading coefficient and r1 , ..., rn are the roots (counting multiplicity) of the polynomial in some splitting field. It is the square of the Vandermonde polynomial times a2n−2 . n As the discriminant is a symmetric function in the roots, it can also be expressed in terms of the coefficients of the polynomial, since the coefficients are the elementary symmetric polynomials in the roots; such a formula is given below. Expressing the discriminant in terms of the roots makes its key property clear, namely that it vanishes if and only if there is a repeated root, but does not allow it to be calculated without factoring a polynomial, after which the information it provides is redundant (if one has the roots, one can tell if there are any duplicates). Hence the formula in terms of the coefficients allows the nature of the roots to be determined without factoring the polynomial.
55.2 Formulas for low degrees The discriminant of a linear polynomial (degree 1) is rarely considered. If needed, it is commonly defined to be equal to 1 (this is compatible with the usual conventions for the empty product and the determinant of the empty matrix). There is no common convention for the discriminant of a constant polynomial (degree 0). The quadratic polynomial
ax2 + bx + c has discriminant
∆ = b2 − 4ac. The cubic polynomial
ax3 + bx2 + cx + d has discriminant
∆ = b2 c2 − 4ac3 − 4b3 d − 27a2 d2 + 18abcd. The quartic polynomial
ax4 + bx3 + cx2 + dx + e has discriminant
∆ = 256a3 e3 − 192a2 bde2 − 128a2 c2 e2 + 144a2 cd2 e − 27a2 d4 + 144ab2 ce2 − 6ab2 d2 e −80abc2 de + 18abcd3 + 16ac4 e − 4ac3 d2 − 27b4 e2 + 18b3 cde − 4b3 d3 − 4b2 c3 e + b2 c2 d2 . These are homogeneous polynomials in the coefficients, respectively of degree 2, 4 and 6. They are also homogeneous in terms of the roots, of respective degrees 2, 6 and 12. Simpler polynomials have simpler expressions for their discriminants. For example, the monic quadratic polynomial x2 + bx + c has discriminant Δ = b2 − 4c. The monic cubic polynomial without quadratic term x3 + px + q has discriminant Δ = −4p3 − 27q2 . In terms of the roots, these discriminants are homogeneous polynomials of respective degree 2 and 6.
55.3. HOMOGENEITY
189
The zero set of discriminant of the cubic x3 + bx2 + cx + d , i.e. points satisfying b2 c2 –4c3 –4b3 d–27d2 +18bcd=0.
55.3 Homogeneity The discriminant is a homogeneous polynomial in the coefficients; it is also a homogeneous polynomial in the roots. In the coefficients, the discriminant is homogeneous of degree 2n−2; this can be seen two ways. In terms of the rootsand-leading-term formula, multiplying all the coefficients by λ does not change the roots, but multiplies the leading term by λ. In terms of the formula as a determinant of a (2n−1) ×(2n−1) matrix divided by an, the determinant of the matrix is homogeneous of degree 2n−1 in the entries, and dividing by an makes the degree 2n−2; explicitly, multiplying the coefficients by λ multiplies all entries of the matrix by λ, hence multiplies the determinant by λ2n−1 . For a monic polynomial, the discriminant is a polynomial in the roots alone (as the an term is one), and is of degree ( ) terms in the product, each squared. n(n−1) in the roots, as there are n2 = n(n−1) 2 Let us consider the polynomial
190
CHAPTER 55. DISCRIMINANT
The discriminant of the quartic polynomial x4 + cx2 + dx + e . The surface represents point (a,b,c) where the polynomial has a repeated roots, the cuspidal edge correspond to polynomials with a triple root and the self intersection to the polynomials with two different repeated roots.
P = a0 xn + a1 xn−1 + · · · + an . It follows from what precedes that its discriminant is homogeneous of degree 2n−2 in the ai and quasi-homogeneous of weight n(n−1) if each ai is given the weight i. In other words, every monomial ai00 · · · , ainn appearing in the discriminant satisfies the two equations
i0 + i1 + · · · + in = 2n − 2 and
0 i0 + 1 i1 + · · · + n in = n(n − 1) These thus correspond to the partitions of n(n−1) into at 2n−2 (non negative) parts of size at most n This restricts the possible terms in the discriminant. For the quadratic polynomial ax2 + bx + c there are only two possibilities for [i0 , i1 , i2 ], either [1,0,1] or [0,2,0], given the two monomials ac and b2 . For the cubic polynomial ax3 + bx2 + cx + d , these are the partitions of 6 into 4 parts of size at most 3: a2 d2 = aadd : 0 + 0 + 3 + 3
abcd : 0 + 1 + 2 + 3
b3 d = bbbd : 1 + 1 + 1 + 3
b2 c2 = bbcc : 1 + 1 + 2 + 2.
ac3 = accc : 0 + 2 + 2 + 2
55.4. QUADRATIC FORMULA
191
All these five monomials occur effectively in the discriminant. While this approach gives the possible terms, it does not determine the coefficients. Moreover, in general not all possible terms will occur in the discriminant. The first example is for the quartic polynomial ax4 +bx3 +cx2 +dx+e , in which case (i0 , . . . , i4 ) = (0, 1, 4, 1, 0) satisfies 0 + 1 + 4 + 1 + 0 = 6 and 1 · 1 + 2 · 4 + 3 · 1 = 12 , even though the corresponding discriminant does not involve the monomial bc4 d .
55.4 Quadratic formula The quadratic polynomial p(x) = ax2 + bx + c has discriminant
∆ = b2 − 4ac, which is the quantity under the square root sign in the quadratic formula. For real numbers a, b, c, one has: • When Δ > 0, P(x) has two distinct real roots √ √ −b ± ∆ −b ± b2 − 4ac = 2a 2a and its graph crosses the x-axis twice. x1,2 =
• When Δ = 0, P(x) has two coincident real roots b 2a and its graph is tangent to the x-axis. x1 = x2 = −
• When Δ < 0, P(x) has no real roots, and its graph lies strictly above or below the x-axis. The polynomial has two distinct complex roots √ √ −b ± i −∆ −b ± i 4ac − b2 z1,2 = = . 2a 2a An alternative way to understand the discriminant of a quadratic is to use the characterization as “zero if and only if the polynomial has a repeated root”. In that case the polynomial is (x − r)2 = x2 − 2rx + r2 . The coefficients then satisfy (−2r)2 = 4(r2 ), so b2 = 4c, and a monic quadratic has a repeated root if and only if this is the case, in which case the root is r = −b/2. Putting both terms on one side and including a leading coefficient yields b2 − 4ac.
55.5 Discriminant of a polynomial To find the formula for the discriminant of a polynomial in terms of its coefficients, it is easiest to introduce the resultant. Just as the discriminant of a single polynomial is the product of the square of the differences between distinct roots, the resultant of two polynomials is the product of the differences between their roots, and just as the discriminant vanishes if and only if the polynomial has a repeated root, the resultant vanishes if and only if the two polynomials share a root. Since a polynomial p(x) has a repeated root if and only if it shares a root with its derivative p′ (x), the discriminant D(p) and the resultant R(p, p′ ) both have the property that they vanish if and only if p has a repeated root, and they have almost the same degree (the degree of the resultant is one greater than the degree of the discriminant) and thus are equal up to a factor of degree one. The benefit of the resultant is that it can be computed as a determinant, namely as the determinant of the Sylvester matrix, a (2n − 1)×(2n − 1) matrix, whose n first rows contain the coefficients of p and the n − 1 last ones the coefficients of its derivative. The resultant R(p, p′ ) of the general polynomial
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CHAPTER 55. DISCRIMINANT
p(x) = an xn + an−1 xn−1 + an−2 xn−2 + . . . + a1 x + a0 is equal to the determinant of the (2n − 1)×(2n − 1) Sylvester matrix: R(p, p′ ) =
an 0 .. .
an−1 an
an−2 an−1
... an−2
a1 ...
a0 a1
0... a0
... 0...
0 nan 0 .. .
... (n − 1)an−1 nan
0 (n − 2)an−2 (n − 1)an−1
an ... (n − 2)an−2
an−1 a1 ...
an−2 0 a1
... ... 0
a1 ... ...
0
0
...
0
nan
(n − 1)an−1
(n − 2)an−2
...
The discriminant D(p) of p(x) is now given by the formula
1
D(p) = (−1) 2 n(n−1)
1 R(p, p′ ). an
For example, in the case n = 4, the above determinant is a4 0 0 4a4 0 0 0
a3 a4 0 3a3 4a4 0 0
a2 a3 a4 2a2 3a3 4a4 0
a1 a2 a3 1a1 2a2 3a3 4a4
a0 a1 a2 0 1a1 2a2 3a3
0 a0 a1 0 0 1a1 2a2
0 0 a0 0 . 0 0 1a1
The discriminant of the degree 4 polynomial is then obtained from this determinant upon dividing by a4 . In terms of the roots, the discriminant is equal to a2n−2 n
∏
(ri − rj )2
i
where r1 , ..., rn are the complex roots (counting multiplicity) of the polynomial: p(x) = an xn + an−1 xn−1 + . . . + a1 x + a0 = an (x − r1 )(x − r2 ) . . . (x − rn ). This second expression makes it clear that p has a multiple root if and only if the discriminant is zero. (This multiple root can be complex.) The discriminant can be defined for polynomials over arbitrary fields, in exactly the same fashion as above. The product formula involving the roots ri remains valid; the roots have to be taken in some splitting field of the polynomial. The discriminant can even be defined for polynomials over any commutative ring. However, if the ring is not an integral domain, above division of the resultant by an should be replaced by substituting an by 1 in the first column of the matrix.
55.6 Nature of the roots The discriminant gives additional information on the nature of the roots beyond simply whether there are any repeated roots: for polynomials with real coefficients, it also gives information on whether the roots are real or complex. This is most transparent and easily stated for quadratic and cubic polynomials; for polynomials of degree 4 or higher this is more difficult to state.
0 0 .. . a0 . 0 0 .. . a1
55.7. DISCRIMINANT OF A POLYNOMIAL OVER A COMMUTATIVE RING
55.6.1
193
Quadratic
Because the quadratic formula expressed the roots of a quadratic polynomial as a rational function in terms of the square root of the discriminant, the roots of a quadratic polynomial are in the same field as the coefficients if and only if the discriminant is a square in the field of coefficients: in other words, the polynomial factors over the field of coefficients if and only if the discriminant is a square. As a real number has real square roots if and only if it is nonnegative, and these roots are distinct if and only if it is positive (not zero), the sign of the discriminant allows a complete description of the nature of the roots of a quadratic polynomial with real coefficients: [5] • Δ > 0: 2 distinct real roots: factors over the reals; • Δ < 0: 2 distinct complex roots (complex conjugate), does not factor over the reals; • Δ = 0: 1 real root with multiplicity 2: factors over the reals as a square. Further, for a quadratic polynomial with rational coefficients, it factors over the rationals if and only if the discriminant – which is necessarily a rational number, being a polynomial in the coefficients – is in fact a square.
55.6.2
Cubic
For more details on this topic, see Cubic polynomial § The nature of the roots. For a cubic polynomial with real coefficients, the discriminant reflects the nature of the roots as follows:
[6]
• Δ > 0: the equation has 3 distinct real roots; • Δ < 0, the equation has 1 real root and 2 complex conjugate roots; • Δ = 0: at least 2 roots coincide, and they are all real. It may be that the equation has a double real root and another distinct single real root; alternatively, all three roots coincide yielding a triple real root. If a cubic polynomial has a triple root, it is a root of its derivative and of its second derivative, which is linear. Thus to decide if a cubic polynomial has a triple root or not, one may compute the root of the second derivative and look if it is a root of the cubic and of its derivative.
55.6.3
Higher degrees
More generally, for a polynomial of degree n with real coefficients, we have • Δ > 0: for some integer k such that 0 ≤ k ≤ roots, all different;
n 4
• Δ < 0: for some integer k such that 0 ≤ k ≤ n−4k−2 real roots, all different;
, there are 2k pairs of complex conjugate roots and n−4k real n−2 4
, there are 2k+1 pairs of complex conjugate roots and
• Δ = 0: at least 2 roots coincide, which may be either real or not real (in this case their complex conjugates also coincide).
55.7 Discriminant of a polynomial over a commutative ring The definition of the discriminant of a polynomial in terms of the resultant may easily be extended to polynomials whose coefficients belong to any commutative ring. However, as the division is not always defined in such a ring, instead of dividing the determinant by the leading coefficient, one substitutes the leading coefficient by 1 in the
194
CHAPTER 55. DISCRIMINANT
first column of the determinant. This generalized discriminant has the following property which is fundamental in algebraic geometry. Let f be a polynomial with coefficients in a commutative ring A and D its discriminant. Let φ be a ring homomorphism of A into a field K and φ(f) be the polynomial over K obtained by replacing the coefficients of f by their images by φ. Then φ(D) = 0 if and only if either the difference of the degrees of f and φ(f) is at least 2 or φ(f) has a multiple root in an algebraic closure of K. The first case may be interpreted by saying that φ(f) has a multiple root at infinity. The typical situation where this property is applied is when A is a (univariate or multivariate) polynomial ring over a field k and φ is the substitution of the indeterminates in A by elements of a field extension K of k. For example, let f be a bivariate polynomial in X and Y with real coefficients, such that f = 0 is the implicit equation of a plane algebraic curve. Viewing f as a univariate polynomial in Y with coefficients depending on X, then the discriminant is a polynomial in X whose roots are the X-coordinates of the singular points, of the points with a tangent parallel to the Y-axis and of some of the asymptotes parallel to the Y-axis. In other words the computation of the roots of the Y-discriminant and the X-discriminant allows to compute all remarkable points of the curve, except the inflection points.
55.8 Generalizations The concept of discriminant has been generalized to other algebraic structures besides polynomials of one variable, including conic sections, quadratic forms, and algebraic number fields. Discriminants in algebraic number theory are closely related, and contain information about ramification. In fact, the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.
55.8.1
Discriminant of a conic section
For a conic section defined in plane geometry by the real polynomial
Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0, the discriminant is equal to[7]
B 2 − 4AC, and determines the shape of the conic section. If the discriminant is less than 0, the equation is of an ellipse or a circle. If the discriminant equals 0, the equation is that of a parabola. If the discriminant is greater than 0, the equation is that of a hyperbola. This formula will not work for degenerate cases (when the polynomial factors).
55.8.2
Discriminant of a quadratic form
There is a substantive generalization to quadratic forms Q over any field K of characteristic ≠ 2. For characteristic 2, the corresponding invariant is the Arf invariant. Given a quadratic form Q, the discriminant or determinant is the determinant of a symmetric matrix S for Q.[8] Change of variables by a matrix A changes the matrix of the symmetric form by AT SA, which has determinant (det A)2 det S, so under change of variables, the discriminant changes by a non-zero square, and thus the class of the discriminant is well-defined in K/(K * )2 , i.e., up to non-zero squares. See also quadratic residue. Less intrinsically, by a theorem of Jacobi, quadratic forms on K n can be expressed, after a linear change of variables, in diagonal form as
a1 x21 + · · · + an x2n . More precisely, a quadratic forms on V may be expressed as a sum
55.9. ALTERNATING POLYNOMIALS
n ∑
195
ai L2i
i=1
where the Li are independent linear forms and n is the number of the variables (some of the ai may be zero). Then the discriminant is the product of the ai, which is well-defined as a class in K/(K * )2 . For K=R, the real numbers, (R* )2 is the positive real numbers (any positive number is a square of a non-zero number), and thus the quotient R/(R* )2 has three elements: positive, zero, and negative. This is a cruder invariant than signature (n0 , n₊, n₋), where n0 is the number 0s and n± is the number of ±1s in diagonal form. The discriminant is then zero if the form is degenerate ( n0 > 0 ), and otherwise it is the parity of the number of negative coefficients, (−1)n− . For K=C, the complex numbers, (C* )2 is the non-zero complex numbers (any complex number is a square), and thus the quotient C/(C* )2 has two elements: non-zero and zero. This definition generalizes the discriminant of a quadratic polynomial, as the polynomial ax2 + bx + c homogenizes to the quadratic form ax2 + bxy + cy 2 which has symmetric matrix [
] a b/2 . b/2 c
whose determinant is ac − (b/2)2 = ac − b2 /4 Up to a factor of −4, this is b2 − 4ac The invariance of the class of the discriminant of a real form (positive, zero, or negative) corresponds to the corresponding conic section being an ellipse, parabola, or hyperbola.
55.8.3
Discriminant of an algebraic number field
Main article: Discriminant of an algebraic number field
55.9 Alternating polynomials Main article: Alternating polynomials The discriminant is a symmetric polynomial in the roots; if one adjoins a square root of it (halves each of the powers: the Vandermonde polynomial) to the ring of symmetric polynomials in n variables Λn , one obtains the ring of alternating polynomials, which is thus a quadratic extension of Λn .
55.10 References [1] Wang, Dongming (2004). Elimination practice: software tools and applications. Imperial College Press. p. 180. ISBN 1-86094-438-8., Chapter 10 page 180 [2] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. (1994). Discriminants, resultants and multidimensional determinants. Birkhäuser. p. 1. ISBN 3-7643-3660-9., Preview page 1 [3] Dickenstein, Alicia; Emiris, Ioannis Z. (2005). Solving polynomial equations: foundations, algorithms, and applications. Springer. p. 26. ISBN 3-540-24326-7., Chapter 1 page 26 [4] J. J. Sylvester (1851) “On a remarkable discovery in the theory of canonical forms and of hyperdeterminants,” Philosophical Magazine, 4th series, 2 : 391-410; Sylvester coins the word “discriminant” on page 406. [5] Irving, Ronald S. (2004), Integers, polynomials, and rings, Springer-Verlag New York, Inc., ISBN 0-387-40397-3, Chapter 10.3 pp. 153–154 [6] Irving, Ronald S. (2004), Integers, polynomials, and rings, Springer-Verlag New York, Inc., ISBN 0-387-40397-3, Chapter 10 ex 10.14.4 and 10.17.4, pp. 154–156
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[7] Fanchi, John R. (2006), Math refresher for scientists and engineers, John Wiley and Sons, pp. 44–45, ISBN 0-471-75715-2, Section 3.2, page 45 [8] Cassels, J.W.S. (1978). Rational Quadratic Forms. London Mathematical Society Monographs 13. Academic Press. p. 6. ISBN 0-12-163260-1. Zbl 0395.10029.
55.11 External links • Mathworld article • Planetmath article
Chapter 56
Discriminant of an algebraic number field “Brill’s theorem” redirects here. For the result in algebraic geometry, see Brill–Noether theorem. In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking,
A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x3 − x2 − 2x + 1. This fundamental domain sits inside K QR. The discriminant of K is 49 = 72 . Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7.
measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified. The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula 197
198
CHAPTER 56. DISCRIMINANT OF AN ALGEBRAIC NUMBER FIELD
for K. An old theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.[1] The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K.
56.1 Definition Let K be an algebraic number field, and let OK be its ring of integers. Let b1 , ..., bn be an integral basis of OK (i.e. a basis as a Z-module), and let {σ1 , ..., σn} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is σi(bj). Symbolically,
σ1 (b1 )
σ2 (b1 ) ∆K = det .. . σn (b1 )
σ1 (b2 ) .. .
··· ..
···
. ···
σ1 (bn ) .. . .. . σn (bn )
2 .
Equivalently, the trace from K to Q can be used. Specifically, define the trace form to be the matrix whose (i,j)-entry is TrK/Q(bibj). This matrix equals BT B, so the discriminant of K is the determinant of this matrix.
56.2 Examples √ • Quadratic number fields: let d be a square-free integer, then the discriminant of K = Q( d) is[2] { ∆K =
d 4d
ifd ≡ 1 (mod 4) ifd ≡ 2, 3 (mod 4).
An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant.[3] • Cyclotomic fields: let n > 2 be an integer, let ζn be a primitive nth root of unity, and let Kn = Q(ζn) be the nth cyclotomic field. The discriminant of Kn is given by[4][2]
∆Kn = (−1)φ(n)/2 ∏
nφ(n) pφ(n)/(p−1)
p|n
where φ(n) is Euler’s totient function, and the product in the denominator is over primes p dividing n. • Power bases: In the case where the ring of integers has a power integral basis, that is, can be written as OK = Z[α], the discriminant of K is equal to the discriminant of the minimal polynomial of α. To see this, one can chose the integral basis of OK to be b1 = 1, b2 = α, b3 = α2 , ..., bn = αn−1 . Then, the matrix in the definition is the Vandermonde matrix associated to αi = σi(α), whose determinant squared is ∏
(αi − αj )2
1≤i
which is exactly the definition of the discriminant of the minimal polynomial.
56.3. BASIC RESULTS
199
• Let K = Q(α) be the number field obtained by adjoining a root α of the polynomial x3 − x2 − 2x − 8. This is Richard Dedekind's original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by {1, α, α(α + 1)/2} and the discriminant of K is −503.[5][6] • Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969. They are obtained by adjoining a root of the polynomial x3 − 21x + 28 or x3 − 21x − 35, respectively.[7]
56.3 Basic results • Brill’s theorem:[8] The sign of the discriminant is (−1)r2 where r2 is the number of complex places of K.[9] • A prime p ramifies in K if, and only if, p divides ΔK .[10] • Stickelberger’s theorem:[11]
∆K ≡ 0 or 1 (mod 4). • Minkowski’s bound:[12] Let n denote the degree of the extension K/Q and r2 the number of complex places of K, then
|∆K |1/2 ≥
n n ( π )r 2 nn ( π )n/2 ≥ . n! 4 n! 4
• Minkowski’s theorem:[13] If K is not Q, then |ΔK| > 1 (this follows directly from the Minkowski bound). • Hermite–Minkowski theorem:[14] Let N be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields K with |ΔK| < N. Again, this follows from the Minkowski bound together Hermite’s theorem (There are only finitely many algebraic number fields with prescribed discriminant).
56.4 History The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871.[15] At this point, he already knew the relationship between the discriminant and ramification.[16] Hermite’s theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857.[17] In 1877, Alexander von Brill determined the sign of the discriminant.[18] Leopold Kronecker first stated Minkowski’s theorem in 1882,[19] though the first proof was given by Hermann Minkowski in 1891.[20] In the same year, Minkowski published his bound on the discriminant.[21] Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four.[22][23]
56.5 Relative discriminant The discriminant defined above is sometimes referred to as the absolute discriminant of K to distinguish it from the relative discriminant ΔK/L of an extension of number fields K/L, which is an ideal in OL. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in OL may not be principal and that there may not be an OL basis of OK. Let {σ1 , ..., σn} be the set of embeddings of K into C which are the identity on L. If b1 , ..., bn is any basis of K over L, let d(b1 , ..., bn) be the square of the determinant of the n by n matrix whose (i,j)-entry is σi(bj). Then, the relative discriminant of K/L is the ideal generated by the d(b1 , ..., bn) as {b1 , ..., bn} varies over all integral bases of K/L. (i.e. bases with the property that bi ∈ OK for all i.) Alternatively, the relative discriminant of K/L is the norm of the different of K/L.[24] When L = Q, the relative
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CHAPTER 56. DISCRIMINANT OF AN ALGEBRAIC NUMBER FIELD
Richard Dedekind showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.[15]
discriminant ΔK/Q is the principal ideal of Z generated by the absolute discriminant ΔK . In a tower of fields K/L/F the relative discriminants are related by
( ) [K:L] ∆K/F = NL/F ∆K/L ∆L/F where N denotes relative norm.[25]
56.6. ROOT DISCRIMINANT
56.5.1
201
Ramification
The relative discriminant regulates the ramification data of the field extension K/L. A prime ideal p of L ramifies in K if, and only if, it divides the relative discriminant ΔK/L. An extension is unramified if, and only if, the discriminant is the unit ideal.[24] The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions, for example, for any field with class number greater than one, its Hilbert class field is a non-trivial unramified extension.
56.6 Root discriminant The root discriminant of a number field, K, of degree n, often denoted rdK, is defined as the n-th root of the absolute value of the (absolute) discriminant of K.[26] The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension. The existence of a class field tower provides bounds on the root discriminant: the existence of an infinite class field tower over Q(√-m) where m = 3·5·7·11·19 shows that there are infinitely many fields with root discriminant 2√m ≈ 296.276.[27] If we let r and 2s be the number of real and complex embeddings, so that n = r + 2s, put ρ = r/n and σ = 2s/n. Set α(ρ, σ) to be the infimum of rdK for K with (r', 2s’) = (ρn, σn). We have (for all n large enough) [27]
α(ρ, σ) ≥ 60.8ρ 22.3σ and on the assumption of the generalized Riemann hypothesis
α(ρ, σ) ≥ 215.3ρ 44.7σ . So we have α(0,1) < 296.276. Martinet has shown α(0,1) < 93 and α(1,0) < 1059.[27][28] Voight 2008 proves that for totally real fields, the root discriminant is > 14, with 1229 exceptions.
56.7 Relation to other quantities √ • When embedded into K ⊗Q R , the volume of the √ fundamental domain of OK is |∆K | (sometimes a different measure is used and the volume obtained is 2−r2 |∆K | , where r2 is the number of complex places of K). • Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of K, and hence in the analytic class number formula, and the Brauer–Siegel theorem. • The relative discriminant of K/L is the Artin conductor of the regular representation of the Galois group of K/L. This provides a relation to the Artin conductors of the characters of the Galois group of K/L, called the conductor-discriminant formula.[29]
56.8 Notes [1] Cohen, Diaz y Diaz & Olivier 2002 [2] Manin, Yu. I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences 49 (Second ed.), p. 130, ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002 [3] Definition 5.1.2 of Cohen 1993 [4] Proposition 2.7 of Washington 1997 [5] Dedekind 1878, pp. 30–31 [6] Narkiewicz 2004, p. 64 [7] Cohen 1993, Theorem 6.4.6
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[8] Koch 1997, p. 11 [9] Lemma 2.2 of Washington 1997 [10] Corollary III.2.12 of Neukirch 1999 [11] Exercise I.2.7 of Neukirch 1999 [12] Proposition III.2.14 of Neukirch 1999 [13] Theorem III.2.17 of Neukirch 1999 [14] Theorem III.2.16 of Neukirch 1999 [15] Dedekind’s supplement X of the second edition of Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie (Dedekind 1871) [16] Bourbaki 1994 [17] Hermite 1857. [18] Brill 1877. [19] Kronecker 1882. [20] Minkowski 1891a. [21] Minkowski 1891b. [22] Stickelberger 1897. [23] All facts in this paragraph can be found in Narkiewicz 2004, pp. 59, 81 [24] Neukirch 1999, §III.2 [25] Corollary III.2.10 of Neukirch 1999 or Proposition III.2.15 of Fröhlich & Taylor 1993 [26] Voight 2008 [27] Koch 1997, pp. 181–182 [28] Martinet, Jacques (1978). “Tours de corps de classes et estimations de discriminants”. Inventiones Mathematicae (in French) 44: 65–73. doi:10.1007/bf01389902. Zbl 0369.12007. [29] Section 4.4 of Serre 1967
56.9 References 56.9.1
Primary sources
• Brill, Alexander von (1877), “Ueber die Discriminante”, Mathematische Annalen 12 (1): 87–89, doi:10.1007/BF01442468, JFM 09.0059.02, MR 1509928, retrieved 2009-08-22 • Dedekind, Richard (1871), Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet (2 ed.), Vieweg, retrieved 2009-08-05 • Dedekind, Richard (1878), "Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen”, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 23 (1), retrieved 2009-08-20 • Hermite, Charles (1857), “Extrait d'une lettre de M. C. Hermite à M. Borchardt sur le nombre limité d'irrationalités auxquelles se réduisent les racines des équations à coefficients entiers complexes d'un degré et d'un discriminant donnés”, Crelle’s Journal 53: 182–192, doi:10.1515/crll.1857.53.182, retrieved 2009-08-20 • Kronecker, Leopold (1882), “Grundzüge einer arithmetischen Theorie der algebraischen Grössen”, Crelle’s journal 92: 1–122, JFM 14.0038.02, retrieved 2009-08-20
56.10. FURTHER READING
203
• Minkowski, Hermann (1891a), “Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen”, Crelle’s journal 107: 278–297, JFM 23.0212.01, retrieved 2009-08-20 • Minkowski, Hermann (1891b), “Théorèmes d'arithmétiques”, Comptes rendus de l'Académie des sciences 112: 209–212, JFM 23.0214.01 • Stickelberger, Ludwig (1897), "Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper”, Proceedings of the First International Congress of Mathematicians, Zürich, pp. 182–193, JFM 29.0172.03
56.9.2
Secondary sources
• Bourbaki, Nicolas (1994), Elements of the history of mathematics, Berlin: Springer-Verlag, ISBN 978-3-54064767-6, MR 1290116. Translated from the original French by John Meldrum • Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Berlin, New York: Springer-Verlag, ISBN 978-3-540-55640-4, MR 1228206 • Cohen, Henri; Diaz y Diaz, Francisco; Olivier, Michel (2002), “A Survey of Discriminant Counting”, in Fieker, Claus; Kohel, David R., Algorithmic Number Theory, Proceedings, 5th International Syposium, ANTS-V, University of Sydney, July 2002, Lecture Notes in Computer Science 2369, Berlin: Springer-Verlag, pp. 80–94, doi:10.1007/3-540-45455-1_7, ISBN 978-3-540-43863-2, ISSN 0302-9743, MR 2041075, retrieved 200908-19 • Fröhlich, Albrecht; Taylor, Martin (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, ISBN 978-0-521-43834-6, MR 1215934 • Koch, Helmut (1997), Algebraic Number Theory, Encycl. Math. Sci. 62 (2nd printing of 1st ed.), SpringerVerlag, ISBN 3-540-63003-1, Zbl 0819.11044 • Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag, ISBN 978-3-540-21902-6, MR 2078267 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Serre, Jean-Pierre (1967), “Local class field theory”, in Cassels, J. W. S.; Fröhlich, Albrecht, Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965, London: Academic Press, ISBN 0-12-163251-2, MR 0220701 • Voight, John (2008), “Enumeration of totally real number fields of bounded root discriminant”, in van der Poorten, Alfred J.; Stein, Andreas, Algorithmic number theory. Proceedings, 8th International Symposium, ANTS-VIII, Banff, Canada, May 2008, Lecture Notes in Computer Science 5011, Berlin: Springer-Verlag, pp. 268–281, arXiv:0802.0194, doi:10.1007/978-3-540-79456-1_18, ISBN 978-3-540-79455-4, MR 2467853, Zbl 1205.11125 • Washington, Lawrence (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83 (2 nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4, MR 1421575, Zbl 0966.11047
56.10 Further reading • Milne, James S. (1998), Algebraic Number Theory, retrieved 2008-08-20
Chapter 57
Drinfeld module In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a “Frobenius twist” of the bundle with a “modification” of it. Drinfeld modules were introduced by Drinfeld (1974), who used them to prove the Langlands conjectures for GL2 of an algebraic function field in some special cases. He later invented shtukas and used shtukas of rank 2 to prove the remaining cases of the Langlands conjectures for GL2 . Laurent Lafforgue proved the Langlands conjectures for GLn of a function field by studying the moduli stack of shtukas of rank n. “Shtuka” is a Russian word штука meaning “a single copy”, which comes from the German noun “Stück”, meaning “piece, item, or unit”. In Russian, the word “shtuka” is also used in slang for a thing with known properties, but having no name in a speaker’s mind.
57.1 Drinfeld modules 57.1.1
The ring of additive polynomials
We let L be a field of characteristic p>0. The ring L{τ} is defined to be the ring of noncommutative (or twisted) polynomials a0 + a1 τ + a2 τ 2 + · · · over L, with the multiplication given by
τ a = ap τ for a∈ L. The element τ can be thought of as a Frobenius element: in fact, L is a left module over L{τ}, with elements of L acting as multiplication and τ acting as the Frobenius endomorphism of L. The ring L{τ} can also be thought of as the ring of all (absolutely) additive polynomials
2
a0 x1 + a1 xp + a2 xp + · · · = a0 τ 0 + a1 τ + a2 τ 2 + · · · in L[x], where a polynomial f is called additive if f(x + y) = f(x) + f(y) (as elements of L[x,y]). The ring of additive polynomials is generated as an algebra over L by the polynomial τ = xp . The multiplication in the ring of additive polynomials is given by composition of polynomials, not by multiplication of commutative polynomials, and is not commutative.
57.1.2
Definition of Drinfeld modules
Let F be an algebraic function field with a finite field of constants and fix a place ∞ of F. Define A to be the ring of elements in F that are regular at every place except possibly ∞ . In particular, A is a Dedekind domain and it is 204
57.2. SHTUKAS
205
discrete in F (with the topology induced by ∞ ). For example we may take A to be the polynomial ring Fq [t] . Let L be a field equipped with a ring homomorphism ι : A → L . A Drinfeld A-module over L is a ring homomorphism ϕ : A → L{τ } whose image is not contained in L, such that the composition of ϕ with d : L{τ } → L, a0 + a1 τ + · · · 7→ a0 coincides with ι : A → L . The condition that the image of A is not in L is a non-degeneracy condition, put in to eliminate trivial cases, while the condition that d ◦ ϕ = ι gives the impression that a Drinfeld module is simply a deformation of the map ϕ . As L{τ} can be thought of as endomorphisms of the additive group of L, a Drinfeld A-module can be regarded as an action of A on the additive group of L, or in other words as an A-module whose underlying additive group is the additive group of L.
57.1.3
Examples of Drinfeld modules
• Define A to be Fp[T], the usual (commutative!) ring of polynomials over the finite field of order p. In other words A is the coordinate ring of an affine genus 0 curve. Then a Drinfeld module ψ is determined by the image ψ(T) of T, which can be any non-constant element of L{τ}. So Drinfeld modules can be identified with non-constant elements of L{τ}. (In the higher genus case the description of Drinfeld modules is more complicated.) • The Carlitz module is the Drinfeld module ψ given by ψ(T) = T+τ, where A is Fp[T] and L is a suitable complete algebraically closed field containing A. It was described by L. Carlitz in 1935, many years before the general definition of Drinfeld module. See chapter 3 of Goss’s book for more information about the Carlitz module. See also Carlitz exponential.
57.2 Shtukas Suppose that X is a curve over the finite field Fp. A (right) shtuka of rank r over a scheme (or stack) U is given by the following data: • Locally free sheaves E, E′ of rank r over U×X together with injective morphisms E → E′ ← (Fr×1)* E, whose cokernels are supported on certain graphs of morphisms from U to X (called the zero and pole of the shtuka, and usually denoted by 0 and ∞), and are locally free of rank 1 on their supports. Here (Fr×1)* E is the pullback of E by the Frobenius endomorphism of U. A left shtuka is defined in the same way except that the direction of the morphisms is reversed. If the pole and zero of the shtuka are disjoint then left shtukas and right shtukas are essentially the same. By varying U, we get an algebraic stack Shtukar of shtukas of rank r, a “universal” shtuka over Shtukar ×X and a morphism (∞,0) from Shtukar to X×X which is smooth and of relative dimension 2r − 2. The stack Shtukar is not of finite type for r > 1. Drinfeld modules are in some sense special kinds of shtukas. (This is not at all obvious from the definitions.) More precisely, Drinfeld showed how to construct a shtuka from a Drinfeld module. See Drinfeld, V. G. Commutative subrings of certain noncommutative rings. Funkcional. Anal. i Prilovzen. 11 (1977), no. 1, 11–14, 96. for details.
57.3 Applications Main article: Lafforgue’s theorem
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The Langlands conjectures for functions fields state (very roughly) that there is a bijection between cuspidal automorphic representations of GLn and certain representations of a Galois group. Drinfeld used Drinfeld modules to prove some special cases of the Langlands conjectures, and later proved the full Langlands conjectures for GL2 by generalizing Drinfeld modules to shtukas. The “hard” part of proving these conjectures is to construct Galois representations with certain properties, and Drinfeld constructed the necessary Galois representations by finding them inside the l-adic cohomology of certain moduli spaces of rank 2 shtukas. Drinfeld suggested that moduli spaces of shtukas of rank r could be used in a similar way to prove the Langlands conjectures for GLr; the formidable technical problems involved in carrying out this program were solved by Lafforgue after many years of effort.
57.4 References 57.4.1
Drinfeld modules
• Drinfeld, V. (1974), “Elliptic modules”, Matematicheskii Sbornik (Russian) 94. English translation in Math. USSR Sbornik 23 (1974) 561–592. • Goss, D. (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 35, Berlin, New York: Springer-Verlag, ISBN 978-3-540-61087-8, MR 1423131 • Gekeler, E.-U. (2001), “Drinfeld module”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4. • Laumon, G. (1996), Cohomology of Drinfeld modular varieties I, II, Cambridge University Press. • Rosen, Michael (2002), “13. Drinfeld modules: an introduction”, Number theory in function fields, Graduate Texts in Mathematics 210, New York, NY: Springer-Verlag, ISBN 0-387-95335-3, Zbl 1043.11079.
57.4.2
Shtukas
• Drinfeld, V. G. Cohomology of compactified moduli varieties of F-sheaves of rank 2. (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 162 (1987), Avtomorfn. Funkts. i Teor. Chisel. III, 107—158, 189; translation in J. Soviet Math. 46 (1989), no. 2, 1789–1821 • Drinfeld, V. G. Moduli varieties of F-sheaves. (Russian) Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 23—41. English translation: Functional Anal. Appl. 21 (1987), no. 2, 107–122. • D. Goss, What is a shtuka? Notices of the Amer. Math. Soc. Vol. 50 No. 1 (2003) • Kazhdan, David A. (1979), “An introduction to Drinfeld’s Shtuka”, in Borel, Armand; Casselman, W., Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 347–356, ISBN 978-0-8218-1437-6, MR 546623
Chapter 58
Dual basis in a field extension In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. This requires the property that the field trace TrL/K provides a non-degenerate quadratic form over K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect field, and hence in the cases where K is finite, or of characteristic zero. A dual basis is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using a second basis for computations. Consider two bases for elements in a finite field, GF(pm ):
B1 = α0 , α1 , . . . , αm−1 and
B2 = γ0 , γ1 , . . . , γm−1 then B2 can be considered a dual basis of B1 provided
Tr(αi · γj ) =
{ 0, if i ̸= j 1, otherwise
Here the trace of a value in GF(pm ) can be calculated as follows:
Tr(β) =
m−1 ∑
i
βp
i=0
Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with a multiplication by the multiplicative identity (usually 1).
207
Chapter 59
Eisenstein reciprocity In algebraic number theory Eisenstein’s reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by Eisenstein (1850), though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839.[1]
59.1 Background and notation 1
Let m > 1 be an integer, and let Om be the ring of integers of the m-th cyclotomic field Q(ζm ), where ζm = e2πi m is a primitive m-th root of unity. m 2 = 1 are units in Om . (There are other units as well.) , . . . ζm The numbers ζm , ζm
59.1.1
Primary numbers
A number α ∈ Om is called primary[2][3] if it is not a unit, is relatively prime to m , and is congruent to a rational (i.e. in Z ) integer (mod (1 − ζm )2 ). The following lemma[4][5] shows that primary numbers in Om are analogous to positive integers in Z. Suppose that α, β ∈ Om and that both α and β are relatively prime to m. Then c α primary. This integer is unique (mod m). • There is an integer c making ζm
• if α and β are primary then α ± β is primary, provided that α ± β is coprime with m . • if α and β are primary then αβ is primary.> • αm is primary. The significance of 1−ζm which appears in the definition is most easily seen when m = l is a prime. In that case l = (1 − ζl )(1 − ζl2 ) . . . (1 − ζll−1 ). Furthermore, the prime ideal (l) of Z is totally ramified in Q(ζl ) (l) = (1 − ζl )l−1 , and the ideal (1 − ζl ) is prime of degree 1.[6][7]
59.1.2 m-th power residue symbol Main article: Power residue symbol
208
59.2. STATEMENT OF THE THEOREM
209
For α, β ∈ Om , the m-th power residue symbol for Om is either zero or an m-th root of unity: { ( ) ζ where ζ m = 1 α = β m 0
if α and β are relatively prime otherwise.
It is the m-th power version of the classical (quadratic, m = 2) Jacobi symbol (assuming α and β are relatively prime): • If η ∈ Om and α ≡ η m (mod β) then
• If
• If
( ) α β
m
( ) α β
( ) α β
= 1. m
̸= 1 then α is not an m-th power (mod β).
= 1 then α may or may not be an m-th power (mod β). m
59.2 Statement of the theorem Let m ∈ Z be an odd prime and a ∈ Z an integer relatively prime to m. Then
59.2.1 (
First supplement ζm a
59.2.2 (
)
am−1 −1 m
= ζm
m
. [8]
Second supplement 1−ζm a
59.2.3
)
( = m
ζm a
) m−1 2
. [8]
m
Eisenstein reciprocity
Let α ∈ Om be primary (and therefore relatively prime to m ), and assume that α is also relatively prime to a Then (α) a m
=
(a) α m
. [9][8]
59.3 Proof The theorem is a consequence of the Stickelberger relation.[10][11] Weil (1975) gives a historical discussion of some early reciprocity laws, including a proof of Eisenstein’s law using Gauss and Jacobi sums that is based on Eisenstein’s original proof.
59.4 Generalization In 1922 Takagi proved that if K ⊃ Q(ζl ) is an arbitrary algebraic number field containing the l -th roots of unity for a prime l , then Eisenstein’s law for l -th powers holds in K. [12]
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CHAPTER 59. EISENSTEIN RECIPROCITY
59.5 Applications 59.5.1
First case of Fermat’s last theorem
Assume that p is an odd prime, that xp + y p + z p = 0 for pairwise relatively prime integers (i.e. in Z ) x, y, z and that p ∤ xyz. This is the first case of Fermat’s last theorem. (The second case is when p | xyz. ) Eisenstein reciprocity can be used to prove the following theorems (Wieferich 1909)[13][14] Under the above assumptions, 2p−1 ≡ 1 (mod p2 ). The only primes below 6.7×1015 that satisfy this are 1093 and 3511. See Wieferich primes for details and current records. (Mirimanoff 1911)[15] Under the above assumptions 3p−1 ≡ 1 (mod p2 ). Analogous results are true for all primes ≤ 113, but the proof does not use Eisenstein’s law. See Wieferich prime#Connection with Fermat’s last theorem. (Furtwängler 1912)[16][17] Under the above assumptions, for every prime r | x, rp−1 ≡ 1 (mod p2 ). (Furtwängler 1912)[18] Under the above assumptions, for every prime r | (x − y),
rp−1 ≡ 1 (mod p2 ).
(Vandiver)[19] Under the above assumptions, if in addition p > 3, then xp ≡ x, y p ≡ y and z p ≡ z (mod p3 ).
59.5.2
Powers mod most primes
Eisenstein’s law can be used to prove the following theorem (Trost, Ankeny, Rogers).[20] Suppose a ∈ Z and that l ∤ a where l is an odd prime. If xl ≡ a (mod p) is solvable for all but finitely many primes p then a = bl .
59.6 See also • Quadratic reciprocity • Cubic reciprocity • Quartic reciprocity • Artin reciprocity • Wieferich’s criterion • Mirimanoff’s congruence
59.7 Notes [1] Lemmermeyer, p. 392. [2] Ireland & Rosen, ch. 14.2 [3] Lemmermeyer, ch. 11.2, uses the term semi-primary. [4] Ireland & Rosen, lemma in ch. 14.2 (first assertion only) [5] Lemmereyer, lemma 11.6 [6] Ireland & Rosen, prop 13.2.7 [7] Lemmermeyer, prop. 3.1
59.8. REFERENCES
211
[8] Lemmermeyer, thm. 11.9 [9] Ireland & Rosen, ch. 14 thm. 1 [10] Ireland & Rosen, ch. 14.5 [11] Lemmermeyer, ch. 11.2 [12] Lemmermeyer, ch. 11 notes [13] Lemmermeyer, ex. 11.33 [14] Ireland & Rosen, th. 14.5 [15] Lemmermeyer, ex. 11.37 [16] Lemmermeyer, ex. 11.32 [17] Ireland & Rosen, th. 14.6 [18] Lemmermeyer, ex. 11.36 [19] Ireland & Rosen, notes to ch. 14 [20] Ireland & Rosen, ch. 14.6, thm. 4. This is part of a more general theorem: Assume xn ≡ a (mod p) for all but finitely n many primes p. Then i) if 8 ∤ n then a = bn but ii) if 8|n then a = bn or a = 2 2 bn .
59.8 References • Eisenstein, Gotthold (1850), “Beweis der allgemeinsten Reciprocitätsgesetze zwischen reellen und komplexen Zahlen”, Verhandlungen der Königlich Preußische Akademie der Wissenschaften zu Berlin (in German): 189– 198, Reprinted in Mathematische Werke, volume 2, pages 712–721 • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer Science+Business Media, ISBN 0-387-97329-X • Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer Science+Business Media, ISBN 3-540-66957-4 • Weil, André (1975), “La cyclotomie jadis et naguère”, Séminaire Bourbaki, Vol. 1973/1974, 26ème année, Exp. No. 452, Lecture Notes in Math 431, Berlin, New York: Springer-Verlag, pp. 318–338, MR 0432517
Chapter 60
Eisenstein sum Not to be confused with Eisenstein series. In mathematics, an Eisenstein sum is a finite sum depending on a finite field and related to a Gauss sum. Eisenstein sums were introduced by Gotthold Eisenstein (1848), named “Eisenstein sums” by Stickelberger (1890), and rediscovered by Yamamoto (1985), who called them relative Gauss sums.
60.1 Definition The Eisenstein sum is given by
E(χ, α) =
∑
χ(t)
T rF /K t=α
where F is a finite extension of the finite field K, and χ is a character of the multiplicative group of F, and α is an element of K (Lemmermeyer 2000, p. 133).
60.2 References • Berndt, Bruce C.; Evans, Ronald J. (1979), “Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer”, Illinois Journal of Mathematics 23 (3): 374–437, ISSN 0019-2082, MR 537798, Zbl 0393.12029 • Eisenstein, Gotthold (1848), “Zur Theorie der quadratischen Zerfällung der Primzahlen 8n + 3,7n + 2 und 7n + 4”, Journal für Reine und Angewandte Mathematik 37: 97–126, ISSN 0075-4102 • Lemmermeyer, Franz (2000), Reciprocity laws, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66957-9, MR 1761696, Zbl 0949.11002 • Lidl, Rudolf; Niederreiter, Harald (1997), Finite fields, Encyclopedia of Mathematics and Its Applications 20 (2nd ed.), Cambridge University Press, ISBN 0-521-39231-4, Zbl 0866.11069 • Yamamoto, K. (1985), “On congruences arising from relative Gauss sums”, Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984), Singapore: World Sci. Publishing, pp. 423–446, MR 827799, Zbl 0634.12017
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Eisenstein’s criterion In mathematics, Eisenstein’s criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorable into the product of non-constant polynomials with rational coefficients. The result is also known as the Schönemann–Eisenstein theorem; although this name is rarely used nowadays, it was common in the early 20th century.[1][2] Suppose we have the following polynomial with integer coefficients.
Q = an xn + an−1 xn−1 + · · · + a1 x + a0 If there exists a prime number p such that the following three conditions all apply: • p divides each ai for i ≠ n, • p does not divide an, and • p2 does not divide a0 , then Q is irreducible over the rational numbers. It will also be irreducible over the integers, unless all its coefficients have a nontrivial factor in common (in which case Q as integer polynomial will have some prime number, necessarily distinct from p, as an irreducible factor). The latter possibility can be avoided by first making Q primitive, by dividing it by the greatest common divisor of its coefficients (the content of Q). This division does not change whether Q is reducible or not over the rational numbers (see Primitive part–content factorization for details), and will not invalidate the hypotheses of the criterion for p (on the contrary it could make the criterion hold for some prime, even if it did not before the division). This criterion is certainly not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but does allow in certain important particular cases to prove irreducibility with very little effort. In some cases the criterion does not apply directly (for any prime number), but it does apply after transformation of the polynomial, in such a way that irreducibility of the original polynomial can be concluded.
61.1 Examples Consider the polynomial Q = 3x4 + 15x2 + 10. In order for Eisenstein’s criterion to apply for a prime number p it must divide both non-leading coefficients 15 and 10, which means only p = 5 could work, and indeed it does since 5 does not divide the leading coefficient 3, and its square 25 does not divide the constant coefficient 10. One may therefore conclude that Q is irreducible over Q (and since it is primitive, over Z as well). Note that since Q is of degree 4, this conclusion could not have been established by only checking that Q has no rational roots (which eliminates possible factors of degree 1), since a decomposition into two quadratic factors could also be possible. Often Eisenstein’s criterion does not apply for any prime number. It may however be that it applies (for some prime number) to the polynomial obtained after substitution (for some integer a) of x + a for x; the fact that the polynomial 213
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CHAPTER 61. EISENSTEIN’S CRITERION
after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift. For example consider H = x2 + x + 2, in which the coefficient 1 of x is not divisible by any prime, Eisenstein’s criterion does not apply to H. But if one substitutes x + 3 for x in H, one obtains the polynomial x2 + 7x + 14, which satisfies Eisenstein’s criterion for the prime number 7. Since the substitution is an automorphism of the ring Q[x], the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally. In this particular example it would have been simpler to argue that H (being monic of degree 2) could only be reducible if it had an integer root, which it obviously does not; however the general principle of trying substitutions in order to make Eisenstein’s criterion apply is a useful way to broaden its scope. Another possibility to transform a polynomial so as to satisfy the criterion, which may be combined with applying a shift, is reversing the order of its coefficients, provided its constant term is nonzero (without which it would be divisible by x anyway). This is so because such polynomials are reducible in R[x] if and only if they are reducible in R[x, x−1 ] (for any integral domain R), and in that ring the substitution of x−1 for x reverses the order of the coefficients (in a manner symmetric about the constant coefficient, but a following shift in the exponent amounts to multiplication by a unit). As an example 2x5 − 4x2 − 3 satisfies the criterion for p = 2 after reversing its coefficients, and (being primitive) is therefore irreducible in Z[x].
61.1.1
Cyclotomic polynomials
An important class of polynomials whose irreducibility can be established using Eisenstein’s criterion is that of the cyclotomic polynomials for prime numbers p. Such a polynomial is obtained by dividing the polynomial xp − 1 by the linear factor x − 1, corresponding to its obvious root 1 (which is its only rational root if p > 2): xp − 1 = xp−1 + xp−2 + · · · + x + 1. x−1 Here, as in the earlier example of H, the coefficients 1 prevent Eisenstein’s criterion from applying directly. However the polynomial will satisfy the criterion for p after substitution of x + 1 for x: this gives ( ) ( ) ( ) p p p (x + 1)p − 1 = xp−1 + xp−2 + · · · + x+ , x p−1 2 1 all of whose non-leading coefficients are divisible by p by properties of binomial coefficients, and whose constant coefficient equal to p, and therefore not divisible by p2 . An alternative way to arrive at this conclusion is to use the identity (a + b)p = ap + bp which is valid in characteristic p (and which is based on the same properties of binomial coefficients, and gives rise to the Frobenius endomorphism), to compute the reduction modulo p of the quotient of polynomials:
(x + 1)p − 1 xp + 1p − 1 xp ≡ = = xp−1 x x x
(mod p),
which means that the non-leading coefficients of the quotient are all divisible by p; the remaining verification that the constant term of the quotient is p can be done by substituting 1 (instead of x + 1) for x into the expanded form xp−1 + ... + x + 1.
61.2 History The criterion is named after Gotthold Eisenstein. However, Theodor Schönemann was the first to publish a version of the criterion,[1] in 1846 in Crelle’s Journal,[3] which reads in translation That (x − a)n + pF(x) will be irreducible to the modulus p2 when F(x) to the modulus p does not contain a factor x−a.
61.3. BASIC PROOF
215
This formulation already incorporates a shift to a in place of 0; the condition on F(x) means that F(a) is not divisible by p, and so pF(a) is divisible by p but not by p2 . As stated it is not entirely correct in that it makes no assumptions on the degree of the polynomial F(x), so that the polynomial considered need not be of the degree n that its expression suggests; the example x2 + p(x3 + 1) ≡ (x2 + p)(px + 1) mod p2 , shows the conclusion is not valid without such hypothesis. Assuming that the degree of F(x) does not exceed n, the criterion is correct however, and somewhat stronger than the formulation given above, since if (x − a)n + pF(x) is irreducible modulo p2 , it certainly cannot decompose in Z[x] into non-constant factors. Subsequently Eisenstein published a somewhat different version in 1850, also in Crelle’s Journal.[4] This version reads in translation When in a polynomial F(x) in x of arbitrary degree the coefficient of the highest term is 1, and all following coefficients are whole (real, complex) numbers, into which a certain (real resp. complex) prime number m divides, and when furthermore the last coefficient is equal to εm, where ε denotes a number not divisible by m: then it is impossible to bring F(x) into the form ( µ )( ) x + a1 xµ−1 + · · · + aµ xν + b1 xν−1 + · · · + bν where μ, ν ≥ 1, μ + ν = deg(F(x)), and all a and b are whole (real resp. complex) numbers; the equation F(x) = 0 is therefore irreducible. Here “whole real numbers” are ordinary integers and “whole complex numbers” are Gaussian integers; one should similarly interpret “real and complex prime numbers”. The application for which Eisenstein developed his criterion was establishing the irreducibility of certain polynomials with coefficients in the Gaussian integers that arise in the study of the division of the lemniscate into pieces of equal arc-length. Remarkably Schönemann and Eisenstein, once having formulated their respective criteria for irreducibility, both immediately apply it to give an elementary proof of the irreducibility of the cyclotomic polynomials for prime numbers, a result that Gauss had obtained in his Disquisitiones Arithmeticae with a much more complicated proof. In fact, Eisenstein adds in a footnote that the only proof for this irreducibility known to him, other than that of Gauss, is one given by Kronecker in 1845. This shows that he was unaware of two different proofs of this statement that Schönemann had given, one in either part of a two-part article, the second of which being the one based on the criterion cited above; this is all the more surprising given the fact that two pages further Eisenstein actually refers (for a different matter) to the first part of Schönemann’s article. In a note (“Notiz”) that appeared in the following issue of the Journal,[5] Schönemann points this out to Eisenstein, and indicates that the latter’s method is not essentially different from the one he used in the second proof.
61.3 Basic proof To prove the validity of the criterion, suppose Q satisfies the criterion for the prime number p, but that it is nevertheless reducible in Q[x], from which we wish to obtain a contradiction. From Gauss’ lemma it follows that Q is reducible in Z[x] as well, and in fact can be written as the product Q = GH of two non-constant polynomials G, H (in case Q is not primitive, one applies the lemma to the primitive polynomial Q/c (where the integer c is the content of Q) to obtain a decomposition for it, and multiplies c into one of the factors to obtain a decomposition for Q). Now reduce Q = GH modulo p to obtain a decomposition in (Z/pZ)[x]. But by hypothesis this reduction for Q leaves its leading term, of the form axn for a non-zero constant a ∈ Z/pZ, as the only nonzero term. But then necessarily the reductions modulo p of G and H also make all non-leading terms vanish (and cannot make their leading terms vanish), since no other decompositions of axn are possible in (Z/pZ)[x], which is a unique factorization domain. In particular the constant terms of G and H vanish in the reduction, so they are divisible by p, but then the constant term of Q, which is their product, is divisible by p2 , contrary to the hypothesis, and one has a contradiction.
61.4 Advanced explanation Applying the theory of the Newton polygon for the p-adic number field, for an Eisenstein polynomial, we are supposed to take the lower convex envelope of the points (0, 1), (1, v1 ), (2, v2 ), ..., (n − 1, vn₋₁), (n, 0),
216
CHAPTER 61. EISENSTEIN’S CRITERION
where vi is the p-adic valuation of ai (i.e. the highest power of p dividing it). Now the data we are given on the vi for 0 < i < n, namely that they are at least one, is just what we need to conclude that the lower convex envelope is exactly the single line segment from (0, 1) to (n, 0), the slope being −1/n. This tells us that each root of Q has p-adic valuation 1/n and hence that Q is irreducible over the p-adic field (since, for instance, no product of any proper subset of the roots has integer valuation); and a fortiori over the rational number field. This argument is much more complicated than the direct argument by reduction mod p. It does however allow one to see, in terms of algebraic number theory, how frequently Eisenstein’s criterion might apply, after some change of variable; and so limit severely the possible choices of p with respect to which the polynomial could have an Eisenstein translate (that is, become Eisenstein after an additive change of variables as in the case of the p-th cyclotomic polynomial). In fact only primes p ramifying in the extension of Q generated by a root of Q have any chance of working. These can be found in terms of the discriminant of Q. For example, in the case x2 + x + 2 given above, the discriminant is −7 so that 7 is the only prime that has a chance of making it satisfy the criterion. Modulo 7, it becomes (x − 3)2 — a repeated root is inevitable, since the discriminant is 0 mod 7. Therefore the variable shift is actually something predictable. Again, for the cyclotomic polynomial, it becomes (x − 1)p−1 mod p; the discriminant can be shown to be (up to sign) pp−2 , by linear algebra methods. More precisely, only totally ramified primes have a chance of being Eisenstein primes for the polynomial. (In quadratic fields, ramification is always total, so the distinction is not seen in the quadratic case like x2 + x + 2 above.) In fact, Eisenstein polynomials are directly linked to totally ramified primes, as follows: if a field extension of the rationals is generated by the root of a polynomial that is Eisenstein at p then p is totally ramified in the extension, and conversely if p is totally ramified in a number field then the field is generated by the root of an Eisenstein polynomial at p.
61.5 Generalization Given an integral domain D, let
Q=
n ∑
ai x i
i=0
be an element of D[x], the polynomial ring with coefficients in D. Suppose there exists a prime ideal p of D such that • ai ∈ p for each i ≠ n, • an ∉ p, and • a0 ∉ p2 , where p2 is the ideal product of p with itself. Then Q cannot be written as a product of two non-constant polynomials in D[x]. If in addition Q is primitive (i.e., it has no non-trivial constant divisors), then it is irreducible in D[x]. If D is a unique factorization domain with field of fractions F, then by Gauss’s lemma Q is irreducible in F[x], whether or not it is primitive (since constant factors are invertible in F[x]); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of D. The latter statement gives original theorem for D = Z or (in Eisenstein’s formulation) for D = Z[i]. The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo p; the essential point is that a single-term polynomial over the integral domain D/p cannot decompose as a product in which at least one of the factors has more than one term (because in such a product there can be no cancellation in the coefficient either of the highest or the lowest possible degree).
61.6. SEE ALSO
61.5.1
217
Example
After Z, one of the basic examples of an integral domain is the polynomial ring D = k[u] in the variable u over the field k. In this case, the principal ideal generated by u is a prime ideal. Eisenstein’s criterion can then be used to prove the irreducibility of a polynomial such as Q(x) = x3 + ux + u in D[x]. Indeed, u does not divide a3 , u2 does not divide a0 , and u divides a0 , a1 and a2 . This shows that this polynomial satisfies the hypotheses of the generalization of Eisenstein’s criterion for the prime ideal p = (u) since, for a principal ideal (u), being an element of (u) is equivalent to being divisible by u.
61.6 See also • Cohn’s irreducibility criterion
61.7 References [1] David A. Cox, Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first, American Mathematical Monthly 118 Vol 1, January 2011, pp. 3–31. [2] H. L. Dorwart, Irreducibility of polynomials, American Mathematical Monthly 42 Vol 6 (1935), 369–381, doi:10.2307/2301357. [3] Dr. Schönemann, Von denjenigen Moduln, welche Potenzen von Primzahlen sind, Journal für die reine und angewandte Mathematik 32, pp. 93–118. The criterion is formulated on p. 100 [4] G. Eisenstein Über die Irredicibilität une einige andere Eigenschaften der Gleichung von welche der Theilung der ganzen Lemniscate abhängt, Journal für die reine und angewandte Mathematik 39, pp. 160–179. The criterion is formulated on p. 166 [5] Dr. Schönemann, Über einige von Herrn Dr. Eisenstein aufgestellte Lehrsätze, irreductible Congruenzen betreffend (S.182 Bd. 39 dieses Journals), Journal für die reine und angewandte Mathematik 40, p. 185–188. The Notiz is on page 188.
• D.J.H. Garling, A Course in Galois Theory, Cambridge University Press, (1986), ISBN 0-521-31249-3. • Hazewinkel, Michiel, ed. (2001), “Algebraic equation”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4
Chapter 62
Elementary number An elementary number is one formalization of the concept of a closed-form number. The elementary numbers form an algebraically closed field containing the roots of arbitrary equations using field operations, exponentiation, and logarithms.
62.1 References • Richardson, Daniel (1997). “How to recognize zero”. Journal of Symbolic Computation 24 (6): pp. 627–645. doi:10.1006/jsco.1997.0157..
218
Chapter 63
Elliptic Gauss sum In mathematics, an elliptic Gauss sum is an analog of a Gauss sum depending on an elliptic curve with complex mutliplication. The quadratic residue symbol in a Gauss sum is replaced by a higher residue symbol such as a cubic or quartic residue symbol, and the exponential function in a Gauss sum is replaced by an elliptic function. They were introduced by Eisenstein (1850), at least in the lemniscate case when the elliptic curve has complex multiplication by i, but seem to have been forgotten or ignored until the paper (Pinch 1988).
63.1 Example (Lemmermeyer 2000, 9.3) gives the following example of an elliptic Gauss sum, for the case of an elliptic curve with complex multiplication by i.
−
∑ t
χ(t)ϕ
( )(p−1)/m t π
where • The sum is over residues mod P whose representatives are Gaussian integers • n is a positive integer • m is a positive integer dividing 4n • p = 4n+1 is a rational prime congruent to 1 mod 4 • φ(z) = sl((1 – i)ωz) where sl is the sine lemniscate function, an elliptic function. • χ is the mth power residue symbol in K with respect to the prime P of K • K is the field k[ζ] • k is the field Q[i] • ζ is a primitive 4n-th root of 1 • π is a primary prime in the Gaussian integers Z[i] with norm p • P is a prime in the ring of integers of K lying above π with inertia degree 1 219
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CHAPTER 63. ELLIPTIC GAUSS SUM
63.2 References • Asai, Tetsuya (2007), “Elliptic Gauss sums and Hecke L-values at s = 1”, Proceedings of the Symposium on Algebraic Number Theory and Related Topics, RIMS Kôkyûroku Bessatsu, B4, Res. Inst. Math. Sci. (RIMS), Kyoto, pp. 79–121, MR 2402004 • Cassou-Noguès, Ph.; Taylor, M. J. (1991), “Un élément de Stickelberger quadratique”, Journal of Number Theory 37 (3): 307–342, doi:10.1016/S0022-314X(05)80046-0, ISSN 0022-314X, MR 1096447 • Eisenstein, Gotthold (1850), "Über einige allgemeine Eigenschaften der Gleichung, von welcher die Teilung der ganzen Lemniskate abhängt, nebst Anwendungen derselben auf die Zahlentheorie”, Journal für Reine und Angewandte Mathematik 39: 224–287, ISSN 0075-4102, Reprinted in Math. Werke II, 556–619 • Lemmermeyer, Franz (2000), Reciprocity laws, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66957-9, MR 1761696 • Pinch, R. (1988), “Galois module structure of elliptic functions”, in Stephens, Nelson M.; Thorne., M. P., Computers in mathematical research (Cardiff, 1986), Inst. Math. Appl. Conf. Ser. New Ser. 14, Oxford University Press, pp. 69–91, ISBN 978-0-19-853620-8, MR 960495
Chapter 64
Elliptic unit In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. They were introduced by Gilles Robert in 1973, and were used by John Coates and Andrew Wiles in their work on the Birch and Swinnerton-Dyer conjecture. Elliptic units are an analogue for imaginary quadratic fields of cyclotomic units. They form an example of an Euler system. A system of elliptic units may be constructed for an elliptic curve E with complex multiplication by the ring of integers R of an imaginary quadratic field F. For simplicity we assume that F has class number one. Let a be an ideal of R with generator α. For a Weierstrass model of E, define ∏
a−1 Θa = α−12 ∆N E
(x − x(P ))−6 .
aP =0,P ̸=0
where Δ is the discriminant and x is the X-coordinate on the Weierstrass model. The function Θ is independent of the choice of model, and is defined over the field of definition of E. Let b be an ideal of R coprime to a and Q an R-generator of the b-torsion. Then Θₐ(Q) is defined over the ray class field K(b), and if b is not a prime power then Θₐ(Q) is a global unit: if b is a power of a prime p then Θₐ(Q) is a unit away from p. The function Θₐ satisfies a distribution relation for b = (β) coprime to a: ∏
Θa (P + R) = Θa (βP ) .
bQ=0
64.1 See also • Modular unit
64.2 References • Coates, J.H.; Greenberg, R.; Ribet, K.A.; Rubin, K. (1999). Arithmetic Theory of Elliptic Curves. Lecture Notes in Mathematics 1716. Springer-Verlag. ISBN 3-540-66546-3. • Coates, John; Wiles, Andrew (1977). “On the conjecture of Birch and Swinnerton-Dyer”. Inventiones Mathematicae 39 (3): 223–251. doi:10.1007/BF01402975. Zbl 0359.14009. • Kubert, Daniel S.; Lang, Serge (1981). Modular units. Grundlehren der Mathematischen Wissenschaften 244. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90517-4. MR 648603. Zbl 0492.12002. • Robert, Gilles Unités elliptiques. (Elliptic units) Bull. Soc. Math. France, Supp. Mém. No. 36. Bull. Soc. Math. France, Tome 101. Société Mathématique de France, Paris, 1973. 77 pp.
221
Chapter 65
Embedding problem In Galois theory, a branch of mathematics, the embedding problem is a generalization of the inverse Galois problem. Roughly speaking, it asks whether a given Galois extension can be embedded into a Galois extension in such a way that the restriction map between the corresponding Galois groups is given.
65.1 Definition Given a field K and a finite group H, one may pose the following question (the so called inverse Galois problem). Is there a Galois extension F/K with Galois group isomorphic to H. The embedding problem is a generalization of this problem: Let L/K be a Galois extension with Galois group G and let f : H → G be an epimorphism. Is there a Galois extension F/K with Galois group H and an embedding α : L → F fixing K under which the restriction map from the Galois group of F/K to the Galois group of L/K coincides with f? Analogously, an embedding problem for a profinite group F consists of the following data: Two profinite groups H and G and two continuous epimorphisms φ : F → G and f : H → G. The embedding problem is said to be finite if the group H is. A solution (sometimes also called weak solution) of such an embedding problem is a continuous homomorphism γ : F → H such that φ = f γ. If the solution is surjective, it is called a proper solution.
65.2 Properties Finite embedding problems characterize profinite groups. The following theorem gives an illustration for this principle. Theorem. Let F be a countably (topologically) generated profinite group. Then 1. F is projective if and only if any finite embedding problem for F is solvable. 2. F is free of countable rank if and only if any finite embedding problem for F is properly solvable.
65.3 References • Luis Ribes, Introduction to Profinite groups and Galois cohomology (1970), Queen’s Papers in Pure and Appl. Math., no. 24, Queen’s university, Kingstone, Ont. • V. V. Ishkhanov, B. B. Lur'e, D. K. Faddeev, The embedding problem in Galois theory Translations of Mathematical Monographs, vol. 165, American Mathematical Society (1997). • Michael D. Fried and Moshe Jarden, Field arithmetic, second ed., revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer-Verlag, Heidelberg, 2005. 222
65.3. REFERENCES
223
• A. Ledet, Brauer type embedding problems Fields Institute Monographs, no. 21, (2005). • Vahid Shirbisheh, Galois embedding problems with abelian kernels of exponent p VDM Verlag Dr. Müller, ISBN 978-3-639-14067-5, (2009). • Almobaideen Wesam, Qatawneh Mohammad, Sleit Azzam, Salah Imad, Efficient mapping scheme of ring topology onto tree-hypercubes , Journal of Applied Sciences, 2007
Chapter 66
Equally spaced polynomial An equally spaced polynomial (ESP) is a polynomial used in finite fields, specifically GF(2) (binary). An s-ESP of degree sm can be written as: ESP (x) =
∑m i=0
xsi for i = 0, 1, . . . , m
or
ESP (x) = xsm + xs(m−1) + · · · + xs + 1.
66.1 Properties Over GF(2) the ESP has many interesting properties, including: • The Hamming weight of the ESP is m + 1. A 1-ESP is known as an all one polynomial and has additional properties including the above.
66.2 References
224
Chapter 67
Equivariant L-function In algebraic number theory, an equivariant Artin L-function is a function associated to a finite Galois extension of global fields created by packaging together the various Artin L-functions associated with the extension. Each extension has many traditional Artin L-functions associated with it, corresponding to the characters of representations of the Galois group. By contrast, each extension has a unique corresponding equivariant L-function. Equivariant L-functions have become increasingly important as a wide range of conjectures and theorems in number theory have been developed around them. Among these are the Brumer–Stark conjecture, the Coates-Sinnott conjecture, and a recently developed equivariant version of the main conjecture in Iwasawa theory.
67.1 References • Solomon, David (2010). “Equivariant L-functions at s=0 and s=1”. Actes de la conférence “Fonctions L et arithmétique”. Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres 2010. Besançon: Laboratoire de Mathématique de Besançon. pp. 129–156. Zbl 05831937.
225
Chapter 68
Euclidean field This article is about ordered fields. For algebraic number fields whose ring of integers has a Euclidean algorithm, see Norm-Euclidean field. For the class of models in statistical mechanics, see Euclidean field theory. In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for some y in K.
68.1 Properties • Every Euclidean field is an ordered Pythagorean field, but the converse is not true.[1] • If E/F is a finite extension, and E is Euclidean, then so is F. This “going-down theorem” is a consequence of the Diller–Dress theorem.[2]
68.2 Examples • The real numbers R with the usual operations and ordering form a Euclidean field. • The field of real algebraic numbers R ∩ Q is a Euclidean field. • The real constructible numbers, those (signed) lengths which can be constructed from a rational segment by ruler and compass constructions, form a Euclidean field.[3] • The field of hyperreal numbers is a Euclidean field.
68.3 Counterexamples • The rational numbers Q with the usual operations and ordering do not form a Euclidean field. For example, 2 is not a square in Q since the square root of 2 is irrational.[4] By the going-down result above, no algebraic number field can be Euclidean.[2] • The complex numbers C do not form a Euclidean field since they cannot be given the structure of an ordered field.
68.4 Euclidean closure The Euclidean closure of an ordered field K is an extension of K in the quadratic closure of K which is maximal with respect to being an ordered field with an order extending that of K.[5] 226
68.5. REFERENCES
227
68.5 References [1] Martin (1998) p. 89 [2] Lam (2005) p.270 [3] Martin (1998) pp. 35–36 [4] Martin (1998) p. 35 [5] Efrat (2006) p. 177
• Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs 124. Providence, RI: American Mathematical Society. ISBN 0-8218-4041-X. Zbl 1103.12002. • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Martin, George E. (1998). Geometric Constructions. Undergraduate Texts in Mathematics. Springer-Verlag. ISBN 0-387-98276-0. Zbl 0890.51015.
68.6 External links • Euclidean Field at PlanetMath.org.
Chapter 69
Euler system In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and the work of Thaine (1988). Euler systems are named after Leonhard Euler because the factors relating different elements of an Euler system resemble the Euler factors of an Euler product. Euler systems can be used to construct annihilators of ideal class groups or Selmer groups, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some Tate-Shafarevich groups. This led to Karl Rubin's new proof of the main conjecture of Iwasawa theory, considered simpler than the original proof due to Barry Mazur and Andrew Wiles.
69.1 Definition Although there are several definitions of special sorts of Euler system, there seems to be no published definition of an Euler system that covers all known cases. But it is possible to say roughly what an Euler system is, as follows: • An Euler system is given by collection of elements cF. These elements are often indexed by certain number fields F containing some fixed number field K, or by something closely related such as square-free integers. The elements cF are typically elements of some Galois cohomology group such as H1 (F, T) where T is a p-adic representation of the absolute Galois group of K. • The most important condition is that the elements cF and cG for two different fields F ⊆ G are related by a simple formula, such as corG/F (cG ) =
∏ q∈Σ(G/F )
−1 P (Fr−1 q |HomO (T, O(1)); Frq )cF
Here the “Euler factor” P(τ|B;x) is defined to be the element det(1-τx|B) considered as an element of O[x], which when x happens to act on B is not the same as det(1-τx|B) considered as an element of O. • There may be other conditions that the cF have to satisfy, such as congruence conditions. Kazuya Kato refers to the elements in an Euler system as “arithmetic incarnations of zeta” and describes the property of being an Euler system as “an arithmetic reflection of the fact that these incarnations are related to special values of Euler products”.[1]
69.2 Examples 69.2.1
Cyclotomic units
For every square-free positive integer n pick an n-th root ζn of 1, with ζmn = ζmζn for m,n coprime. Then the cyclotomic Euler system is the set of numbers αn = 1 − ζn. These satisfy the relations 228
69.3. NOTES
229
NQ(ζnl )/Q(ζl ) (αnl ) = αnFl −1 αnl ≡ αn modulo all primes above l where l is a prime not dividing n and Fl is a Frobenius automorphism with Fl(ζn) = ζl n. Kolyvagin used this Euler system to give an elementary proof of the Gras conjecture.
69.2.2
Gauss sums
69.2.3
Elliptic units
69.2.4
Heegner points
Kolyvagin constructed an Euler system from the Heegner points of an elliptic curve, and used this to show that in some cases the Tate-Shafarevich group is finite.
69.2.5
Kato’s Euler system
Kato’s Euler system consists of certain elements occurring in the algebraic K-theory of modular curves. These elements—named Beilinson elements after Alexander Beilinson who introduced them in Beilinson (1984)—were used by Kazuya Kato in Kato (2004) to prove one divisibility in Barry Mazur’s main conjecture of Iwasawa theory for elliptic curves.[2]
69.3 Notes [1] Kato 2007, §2.5.1 [2] Kato 2007
69.4 References • Banaszak, Grzegorz (2001), “Euler systems for number fields”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Beilinson, Alexander (1984), “Higher regulators and values of L-functions”, in R. V. Gamkrelidze, Current problems in mathematics (in Russian) 24, pp. 181–238, MR 0760999 • Coates, J.H.; Greenberg, R.; Ribet, K.A.; Rubin, K. (1999), Arithmetic Theory of Elliptic Curves, Lecture Notes in Mathematics 1716, Springer-Verlag, ISBN 3-540-66546-3 • Coates, J.; Sujatha, R. (2006), “Euler systems”, Cyclotomic Fields and Zeta Values, Springer Monographs in Mathematics, Springer-Verlag, pp. 71–87, ISBN 3-540-33068-2 • Kato, Kazuya (2004), "p-adic Hodge theory and values of zeta functions of modular forms”, in Pierre Berthelot, Jean-Marc Fontaine, Luc Illusie, Kazuya Kato, and Michael Rapoport, Cohomologies p-adiques et applications arithmétiques. III., Astérisque 295, Paris: Société Mathématique de France, pp. 117–290, MR 2104361 • Kato, Kazuya (2007), “Iwasawa theory and generalizations”, in Marta Sanz-Solé; Javier Soria; Juan Luis Varona et al., International Congress of Mathematicians (PDF) I, Zürich: European Mathematical Society, pp. 335– 357, MR 2334196, retrieved 2010-08-12 . Proceedings of the congress held in Madrid, August 22–30, 2006 • Kolyvagin, V. A. (1988), “The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves”, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 52 (6): 1154–1180, ISSN 0373-2436, MR 984214 • Kolyvagin, V. A. (1990), “Euler systems”, The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Boston, MA: Birkhäuser Boston, pp. 435–483, doi:10.1007/978-0-8176-4575-5_11, ISBN 978-0-8176-3428-5, MR 1106906
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• Mazur, Barry; Rubin, Karl (2004), “Kolyvagin systems” (PDF), Memoirs of the American Mathematical Society 168 (799): viii+96, doi:10.1090/memo/0799, ISBN 978-0-8218-3512-8, ISSN 0065-9266, MR 2031496 • Rubin, Karl (2000), Euler systems, Annals of Mathematics Studies 147, Princeton University Press, MR 1749177 • Scholl, A. J. (1998), “An introduction to Kato’s Euler systems”, Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge University Press, pp. 379–460, ISBN 978-0-521-64419-8, MR 1696501 • Thaine, Francisco (1988), “On the ideal class groups of real abelian number fields”, Annals of Mathematics. Second Series 128 (1): 1–18, doi:10.2307/1971460, ISSN 0003-486X, MR 951505
69.5 External links • Several papers on Kolyvagin systems are available at Barry Mazur’s web page (as of July 2005).
Chapter 70
Explicit reciprocity law In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field. The name “explicit reciprocity law” refers to the fact that the Hilbert symbols of local fields appear in Hilbert’s reciprocity law for the power residue symbol. The definitions of the Hilbert symbol are usually rather roundabout and can be hard to use directly in explicit examples, and the explicit reciprocity laws give more explicit expressions for the Hilbert symbol that are sometimes easier to use. There are also several explicit reciprocity laws for various generalizations of the Hilbert symbol to higher local fields, p-divisible groups, and so on.
70.1 History Artin & Hasse (1928) gave an explicit formula for the Hilbert symbol (α,β) in the case of odd prime powers, for some special values of α and β when the field is the (cyclotomic) extension of the p-adic numbers by a pn th root of unity. Iwasawa (1968) extended the formula of Artin and Hasse to more cases of α and β, and Wiles (1978) and de Shalit (1986) extended Iwasawa’s work to Lubin–Tate extensions of local fields. Shafarevich (1950) gave an explicit formula for the Hilbert symbol for odd prime powers for general local fields. His formula was rather complicated which made it hard to use, and Brückner (1967, 1979) and Vostokov (1978) found a simpler formula. Henniart (1981) simplified Vostokov’s work and extended it to the case of even prime powers.
70.2 Examples For archimedean local fields or in the unramified case the Hilbert symbol is easy to write down explicitly. The main problem is to evaluate it in the ramified case.
70.2.1
Archimedean fields
Over the complex numbers (a, b) is always 1. Over the reals, the Hilbert symbol of odd degree is trivial, and the Hilbert symbol of even degree is given by (a, b)∞ is +1 if at least one of a or b is positive, and −1 if both are negative.
70.2.2
Unramified case: the tame Hilbert symbol
In the unramified case, when the order of the Hilbert symbol is coprime to the residue characteristic of the local field, the tame Hilbert symbol is given by[1]
(a, b) = ω((−1)ord(a) ord(b) bord(a) /aord(b) )(q−1)/n 231
232
CHAPTER 70. EXPLICIT RECIPROCITY LAW
where ω(a) is the (q – 1)-th root of unity congruent to a and ord(a) is the value of the valuation of the local field, and n is the degree of the Hilbert symbol, and q is the order of the residue class field. The number n divides q – 1 because the local field contains the nth roots of unity by assumption. As a special case, over the p-adics with p odd, writing a = pα u and b = pβ v , where u and v are integers coprime to p, we have for the quadratic Hilbert symbol (a, b)p = (−1)αβε(p)
( )β ( )α u p
v p
, where ε(p) = (p − 1)/2
and the expression involves two Legendre symbols.
70.2.3
Ramified case
The simplest example of a Hilbert symbol in the ramified case is the quadratic Hilbert symbol over the 2-adic integers. Over the 2-adics, again writing a = 2α u and b = 2β v , where u and v are odd numbers, we have for the quadratic Hilbert symbol (a, b)2 = (−1)ϵ(u)ϵ(v)+αω(v)+βω(u) , where ω(x) = (x2 − 1)/8.
70.3 See also • Rational reciprocity law
70.4 Notes [1] Neukirch (1999) p.335
70.5 References • Artin, E.; Hasse, H. (1928), “Die beiden Ergänzungssätze zum Reziprozitätsgesetz der ln -ten Potenzreste im Körper der ln -ten Einheitswurzeln”, Abhandlungen Hamburg 6: 146–162, doi:10.1007/bf02940607, JFM 54.0191.05 • Brückner, Helmut (1967), “Eine explizite Formel zum Reziprozitätsgesetz für Primzahlexponenten p”, Algebraische Zahlentheorie (Ber. Tagung Math. Forschungsinst. Oberwolfach, 1964) (in German), Bibliographisches Institut, Mannheim, pp. 31–39, MR 0230702 • Brückner, H. (1979), Explizites Reziprozitätsgesetz und Anwendungen, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen (in German) 2, Universität Essen, Fachbereich Mathematik, Essen, MR 0533354 • de Shalit, Ehud (1986), “The explicit reciprocity law in local class field theory”, Duke Math. J. 53 (1): 163–176, doi:10.1215/s0012-7094-86-05311-1, MR 0835803 • Henniart, Guy (1981), “Sur les lois de réciprocité explicites. I.”, J. Reine Angew. Math. (in French) 329: 177–203, MR 0636453 • Iwasawa, Kenkichi (1968), “On explicit formulas for the norm residue symbol”, J. Math. Soc. Japan 20: 151–165, doi:10.2969/jmsj/02010151, MR 0229609 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Shafarevich, I. R. (1950), “A general reciprocity law”, Mat. Sbornik N.S. (in Russian) 26: 113–146, MR 0031944
70.6. FURTHER READING
233
• Vostokov, S. V. (1978), “An explicit form of the reciprocity law”, Izv. Akad. Nauk SSSR Ser. Mat. 42 (6): 1288–1321, 1439, doi:10.1070/IM1979v013n03ABEH002077, MR 0522940 • Wiles, A. (1978), “Higher explicit reciprocity laws”, Annals of Mathematics 107 (2): 235–254, doi:10.2307/1971143, MR 0480442
70.6 Further reading • Lemmermeyer, Franz (2000), Reciprocity laws. From Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer-Verlag, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002
Chapter 71
Exponential field In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation.
71.1 Definition A field is an algebraic structure composed of a set of elements, F, two binary operations, addition (+) such that F forms an abelian group with identity 0F and multiplication (·), such that F excluding 0F forms an abelian group under multiplication with identity 1F, and such that multiplication is distributive over addition, that is for any elements a, b, c in F, one has a · (b + c) = (a · b) + (a · c). If there is also a function E that maps F into F, and such that for every a and b in F one has
E(a + b) = E(a) · E(b), E(0F ) = 1F then F is called an exponential field, and the function E is called an exponential function on F.[1] Thus an exponential function on a field is a homomorphism from the additive group of F to its multiplicative group.
71.2 Trivial exponential function There is a trivial exponential function on any field, namely the map that sends every element to the identity element of the field under multiplication. Thus every field is trivially also an exponential field, so the cases of interest to mathematicians occur when the exponential function is non-trivial. Exponential fields are sometimes required to have characteristic zero as the only exponential function on a field with nonzero characteristic is the trivial one.[2] To see this first note that for any element x in a field with characteristic p > 0,
1 = E(0) = E(x + x + . . . + x) = E(x)E(x) · · · E(x) = E(x)p . | {z } pthese of
Hence, taking into account the Frobenius endomorphism,
(E(x) − 1)p = E(x)p − 1p = E(x)p − 1 = 0. And so E(x) = 1 for every x.[3] 234
71.3. EXAMPLES
235
71.3 Examples • The field of real numbers R, or (R, +, ·, 0, 1) as it may be written to highlight that we are considering it purely as a field with addition, multiplication, and special constants zero and one, has infinitely many exponential functions. One such function is the usual exponential function, that is E(x) = ex , since we have ex+y = ex ey and e0 = 1, as required. Considering the ordered field R equipped with this function gives the ordered real exponential field, denoted Rₑₓ = (R, +, ·, <, 0, 1, exp). • Any real number a > 0 gives an exponential function on R, where the map E(x) = ax satisfies the required properties. • Analogously to the real exponential field, there is the complex exponential field, Cₑₓ = (C, +, ·, 0, 1, exp). • Boris Zilber constructed an exponential field Kₑₓ that, crucially, satisfies the equivalent formulation of Schanuel’s conjecture with the field’s exponential function.[4] It is conjectured that this exponential field is actually Cₑₓ , and a proof of this fact would thus prove Schanuel’s conjecture.
71.4 Exponential rings The underlying set F may not be required to be a field but instead allowed to simply be a ring, R, and concurrently the exponential function is relaxed to be a homomorphism from the additive group in R to the multiplicative group of units in R. The resulting object is called an exponential ring.[2] An example of an exponential ring with a nontrivial exponential function is the ring of integers Z equipped with the function E which takes the value +1 at even integers and −1 at odd integers, i.e., the function n 7→ (−1)n . This exponential function, and the trivial one, are the only two functions on Z that satisfy the conditions.[5]
71.5 Open problems Exponential fields are much-studied objects in model theory, occasionally providing a link between it and number theory as in the case of Zilber’s work on Schanuel’s conjecture. It was proved in the 1990s that Rₑₓ is model complete, a result known as Wilkie’s theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that Rₑₓ is also o-minimal.[6] On the other hand it is known that Cₑₓ is not model complete.[7] The question of decidability is still unresolved. Alfred Tarski posed the question of the decidability of Rₑₓ and hence it is now known as Tarski’s exponential function problem. It is known that if the real version of Schanuel’s conjecture is true then Rₑₓ is decidable.[8]
71.6 See also • Ordered exponential field • Exponentially closed field
71.7 Notes [1] Helmut Wolter, Some results about exponential fields (survey), Mémoires de la S.M.F. 2e série, 16, (1984), pp.85–94. [2] Lou van den Dries, Exponential rings, exponential polynomials and exponential functions, Pacific Journal of Mathematics, 113, no.1 (1984), pp.51–66. [3] Martin Bays, Jonathan Kirby, A.J. Wilkie, A Schanuel property for exponentially transcendental powers, (2008), arXiv:0810.4457 [4] Boris Zilber, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Ann. Pure Appl. Logic, 132, no.1 (2005), pp.67–95.
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[5] Giuseppina Terzo, Some Consequences of Schanuel’s Conjecture in Exponential Rings, Communications in Algebra, Volume 36, Issue 3 (2008), pp.1171–1189. [6] A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc., 9 (1996), pp. 1051–1094. [7] David Marker, A remark on Zilber’s pseudoexponentiation, The Journal of Symbolic Logic, 71, no.3 (2006), pp. 791–798. [8] A.J. Macintyre, A.J. Wilkie, On the decidability of the real exponential field, Kreisel 70th Birthday Volume, (2005).
Chapter 72
Exponentially closed field In mathematics, an exponentially closed field is an ordered field F which has an order preserving isomorphism E of the additive group of F onto the multiplicative group of positive elements of F such that 1 + 1/n < E(1) < n for some natural number n . Isomorphism E is called an exponential function in F .
72.1 Examples • The canonical example for an exponentially closed field is the ordered field of real numbers; here E can be any function ax where 1 < a ∈ F .
72.2 Properties • Every exponentially closed field F is root-closed, i.e., every positive element of F has an n -th root for all positive integer words the multiplicative group of positive elements of F is divisible). This is ( n (or in )other n so because E n1 E −1 (a) = E(E −1 (a)) = a for all a > 0 . • Consequently, every exponentially closed field is an Euclidean field. • Consequently, every exponentially closed field is an ordered Pythagorean field. • Not every real-closed field is an exponentially closed field, e.g., the field of real algebraic numbers is not exponentially closed. This is so because E has to be E(x) = ax for some 1 < a ∈ F in every exponentially √ √ closed subfield F of the real numbers; and E( 2) = a 2 is not algebraic if 1 < a is algebraic by Gelfond– Schneider theorem. • Consequently, the class of exponentially closed fields is not an elementary class since the field of real numbers and the field of real algebraic numbers are elementarily equivalent structures. • The class of exponentially closed fields is a pseudoelementary class. This is so since a field F is exponentially closed iff there is a surjective function E2 : F → F + such that E2 (x + y) = E2 (x)E2 (y) and E2 (1) = 2 ; and these properties of E2 are axiomatizable.
72.3 See also • Ordered exponential field • Exponential field 237
238
CHAPTER 72. EXPONENTIALLY CLOSED FIELD
72.4 References Alling, Norman L. (1962). “On Exponentially Closed Fields”. Proceedings of the American Mathematical Society 13 (5): 706–711. doi:10.2307/2034159. Zbl 0136.32201.
Chapter 73
Extension and contraction of ideals In commutative algebra, the extension and contraction of ideals are operations performed on sets of ideals.
73.1 Extension of an ideal Let A and B be two commutative rings with unity, and let f : A → B be a (unital) ring homomorphism. If a is an ideal in A, then f (a) need not be an ideal in B (e.g. take f to be the inclusion of the ring of integers Z into the field of rationals Q). The extension ae of a in B is defined to be the ideal in B generated by f (a) . Explicitly,
ae =
{∑
yi f (xi ) : xi ∈ a, yi ∈ B
}
73.2 Contraction of an ideal If b is an ideal of B, then f −1 (b) is always an ideal of A, called the contraction bc of b to A.
73.3 Properties Assuming f : A → B is a unital ring homomorphism, a is an ideal in A, b is an ideal in B, then: • b is prime in B ⇒ bc is prime in A. • aec ⊇ a • bce ⊆ b •
• It is false, in general, that a being prime (or maximal) in A implies that ae is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding Z → Z [i] . In B = Z [i] , the element 2 factors as 2 = (1 + i)(1 − i) where (one can show) neither of 1 + i, 1 − i are units in B. So (2)e is not prime in B (and therefore not maximal, as well). Indeed, (1 ± i)2 = ±2i shows that (1 + i) = ((1 − i) − (1 − i)2 ) , (1 − i) = ((1 + i) − (1 + i)2 ) , and therefore (2)e = (1 + i)2 .
On the other hand, if f is surjective and a ⊇ kerf then: • aec = a and bce = b . 239
240
CHAPTER 73. EXTENSION AND CONTRACTION OF IDEALS
• Extension of prime ideals is not prime in general, but if f is surjective a is a prime ideal in A ⇔ ae is a prime ideal in B. • a is a maximal ideal in A ⇔ ae is a maximal ideal in B.
73.4 Extension of prime ideals in number theory Let K be a field extension of L, and let B and A be the rings of integers of K and L, respectively. Then B is an integral extension of A, and we let f be the inclusion map from A to B. The behaviour of a prime ideal a = p of A under extension is one of the central problems of algebraic number theory.
73.5 See also • Splitting of prime ideals in Galois extensions
73.6 References • Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-20100361-9
Chapter 74
Field (mathematics) This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, algebraic number fields, p-adic fields, and so forth. Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry. As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example the integers form a ring, but 2x = 1 has no solution in integers. Also, the multiplication operation in a field is required to be commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) is called a division ring or skew field. (Historically, division rings were sometimes referred to as fields, while fields were called commutative fields.) As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following (not exhaustive) chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
74.1 Definition and illustration Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition and multiplication (the latter excluding zero), with “compatible” being formalized by distributivity, and the caveat that the additive and the multiplicative identities are distinct (0 ≠ 1). The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold; subtraction and division are defined in terms of the inverse operations of addition and multiplication:[note 1] Closure of F under addition and multiplication For all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary operations on F). Associativity of addition and multiplication For all a, b, and c in F, the following equalities hold: a + (b + c) = (a 241
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CHAPTER 74. FIELD (MATHEMATICS) + b) + c and a · (b · c) = (a · b) · c.
Commutativity of addition and multiplication For all a and b in F, the following equalities hold: a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element of F, called the additive identity element and denoted by 0, such that for all a in F, a + 0 = a. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 = 1. (The elements a + (−b) and a · b−1 are also denoted a − b and a/b, respectively.) In other words, subtraction and division operations exist. Distributivity of multiplication over addition For all a, b and c in F, the following equality holds: a · (b + c) = (a · b) + (a · c). A field is therefore an algebraic structure F, +, ·, −, −1 , 0, 1 ; of type 2, 2, 1, 1, 0, 0 , consisting of two abelian groups: • F under +, −, and 0; • F ∖ {0} under ·, −1 , and 1, with 0 ≠ 1, with · distributing over +.[1]
74.1.1
First example: rational numbers
A simple example of a field is the field of rational numbers, consisting of numbers which can be written as fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is simply −a/b, and the multiplicative inverse (provided that a ≠ 0) is b/a. To see the latter, note that
b a ba · = = 1. a b ab The abstractly required field axioms reduce to standard properties of rational numbers, such as the law of distributivity
a · b
(
c e + d f (
)
a · b
=
a · b
=
a(cf + ed) acf aed ac ae = + = + bdf bdf bdf bd bf
=
a c a e · + · , b d b f
(
c f e d · + · d f f d
)
=
cf ed + df fd
) =
a cf + ed · b df
or the law of commutativity and law of associativity.
74.2. RELATED ALGEBRAIC STRUCTURES
74.1.2
243
Second example: a field with four elements
In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called O, I, A and B. The notation is chosen such that O plays the role of the additive identity element (denoted 0 in the axioms), and I is the multiplicative identity (denoted 1 above). One can check that all field axioms are satisfied. For example: A · (B + A) = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity. The above field is called a finite field with four elements, and can be denoted F4 . Field theory is concerned with understanding the reasons for the existence of this field, defined in a fairly ad-hoc manner, and describing its inner structure. For example, from a glance at the multiplication table, it can be seen that any non-zero element (i.e., I, A, and B) is a power of A: A = A1 , B = A2 = A · A, and finally I = A3 = A · A · A. This is not a coincidence, but rather one of the starting points of a deeper understanding of (finite) fields.
74.1.3
Alternative axiomatizations
As with other algebraic structures, there exist alternative axiomatizations. Because of the relations between the operations, one can alternatively axiomatize a field by explicitly assuming that there are four binary operations (add, subtract, multiply, divide) with axioms relating these, or (by functional decomposition) in terms of two binary operations (add and multiply) and two unary operations (additive inverse and multiplicative inverse), or other variants. The usual axiomatization in terms of the two operations of addition and multiplication is brief and allows the other operations to be defined in terms of these basic ones, but in other contexts, such as topology and category theory, it is important to include all operations as explicitly given, rather than implicitly defined (compare topological group). This is because without further assumptions, the implicitly defined inverses may not be continuous (in topology), or may not be able to be defined (in category theory). Defining an inverse requires that one is working with a set, not a more general object. For a very economical axiomatization of the field of real numbers, whose primitives are merely a set R with 1 ∈ R, addition, and a binary relation, "<". See Tarski’s axiomatization of the reals.
74.2 Related algebraic structures The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the existence of the binary operation "·", together with its commutativity, associativity, (multiplicative) identity element and inverses are precisely the axioms for an abelian group. In other words, for any field, the subset of nonzero elements F \ {0}, also often denoted F × , is an abelian group (F × , ·) usually called multiplicative group of the field. Likewise (F, +) is an abelian group. The structure of a field is hence the same as specifying such two group structures (on the same set), obeying the distributivity. Important other algebraic structures such as rings arise when requiring only part of the above axioms. For example, if the requirement of commutativity of the multiplication operation · is dropped, one gets structures usually called division rings or skew fields.
74.2.1
Remarks
By elementary group theory, applied to the abelian groups (F × , ·), and (F, +), the additive inverse −a and the multiplicative inverse a−1 are uniquely determined by a. Similar direct consequences from the field axioms include −(a · b) = (−a) · b = a · (−b), in particular −a = (−1) · a as well as a · 0 = 0. Both can be shown by replacing b or c with 0 in the distributive property.
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74.3 History The concept of field was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability of polynomial equations with rational coefficients of degree five or higher. In 1857, Karl von Staudt published his Algebra of Throws which provided a geometric model satisfying the axioms of a field.[2] This construction has been frequently recalled as a contribution to the foundations of mathematics. In 1871, Richard Dedekind introduced, for a set of real or complex numbers which is closed under the four arithmetic operations, the German word Körper, which means “body” or “corpus” (to suggest an organically closed entity),[3] hence the common use of the letter K to denote a field. He also defined rings (then called order or order-modul), but the term “a ring” (Zahlring) was invented by Hilbert.[4] In 1893, Eliakim Hastings Moore called the concept “field” in English.[5][6] In 1881, Leopold Kronecker defined what he called a “domain of rationality”, which is indeed a field of polynomials in modern terms. In 1893, Heinrich M. Weber gave the first clear definition of an abstract field.[7] In 1910, Ernst Steinitz published the very influential paper Algebraische Theorie der Körper (English: Algebraic Theory of Fields).[8] In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like prime field, perfect field and the transcendence degree of a field extension. Emil Artin developed the relationship between groups and fields in great detail from 1928 through 1942.
74.4 Examples 74.4.1
Rationals and algebraic numbers
The field of rational numbers Q has been introduced above. A related class of fields very important in number theory are algebraic number fields. We will first give an example, namely the field Q(ζ) consisting of numbers of the form a + bζ with a, b ∈ Q, where ζ is a primitive third root of unity, i.e., a complex number satisfying ζ3 = 1, ζ ≠ 1. This field extension can be used to prove a special case of Fermat’s last theorem, which asserts the non-existence of rational nonzero solutions to the equation x3 + y3 = z3 . In the language of field extensions detailed below, Q(ζ) is a field extension of degree 2. Algebraic number fields are by definition finite field extensions of Q, that is, fields containing Q having finite dimension as a Q-vector space.
74.4.2
Reals, complex numbers, and p-adic numbers
Take the real numbers R, under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered field; it is this structure which provides the foundation for most formal treatments of calculus. The complex numbers C consist of expressions a + bi where i is the imaginary unit, i.e., a (non-real) number satisfying i2 = −1. Addition and multiplication of real numbers are defined in such a way that all field axioms hold for C. For example, the distributive law enforces (a + bi)·(c + di) = ac + bci + adi + bdi2 , which equals ac−bd + (bc + ad)i. The real numbers can be constructed by completing the rational numbers, i.e., filling the “gaps": for example √2 is such a gap. By a formally very similar procedure, another important class of fields, the field of p-adic numbers Qp is built. It is used in number theory and p-adic analysis.
74.4. EXAMPLES
245
Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers.
74.4.3
Constructible numbers
Given 0, 1, r1 and r2 , the construction yields r1 ·r2
In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compass and straightedge. For example it was unknown to the Greeks that it is in general impossible to trisect a given angle. Using the field notion and field theory allows these problems to be settled. To do so, the field of constructible numbers is considered. It contains, on the plane, the points 0 and 1, and all complex numbers that can be constructed from these two by a finite number of construction steps using only compass and straightedge. This set, endowed with the usual addition and multiplication of complex numbers does form a field. For example, multiplying two (real) numbers r1 and r2 that have already been constructed can be done using construction at the right, based on the intercept theorem. This way, the obtained field F contains all rational numbers, but is bigger than Q, because for any f ∈ F, the square root of f is also a constructible number. A closely related concept is that of a Euclidean field, namely an ordered field whose positive elements are closed under square root. The real constructible numbers form the least Euclidean field, and the Euclidean fields are precisely the ordered extensions thereof.
74.4.4
Finite fields
Main article: Finite field Finite fields (also called Galois fields) are fields with finitely many elements. The above introductory example F4 is a field with four elements. F2 consists of two elements, 0 and 1. This is the smallest field, because by definition a field has at least two distinct elements 1 ≠ 0. Interpreting the addition and multiplication in this latter field as XOR and AND operations, this field finds applications in computer science, especially in cryptography and coding theory. In a finite field there is necessarily an integer n such that 1 + 1 + ··· + 1 (n repeated terms) equals 0. It can be shown that the smallest such n must be a prime number, called the characteristic of the field. If a (necessarily infinite) field
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has the property that 1 + 1 + ··· + 1 is never zero, for any number of summands, such as in Q, for example, the characteristic is said to be zero. A basic class of finite fields are the fields Fp with p elements (p a prime number): Fp = Z/pZ = {0, 1, ..., p − 1}, where the operations are defined by performing the operation in the set of integers Z, dividing by p and taking the remainder; see modular arithmetic. A field K of characteristic p necessarily contains Fp,[9] and therefore may be viewed as a vector space over Fp, of finite dimension if K is finite. Thus a finite field K has prime power order, i.e., K has q = pn elements (where n > 0 is the number of elements in a basis of K over Fp). By developing more field theory, in particular the notion of the splitting field of a polynomial f over a field K, which is the smallest field containing K and all roots of f, one can show that two finite fields with the same number of elements are isomorphic, i.e., there is a one-to-one mapping of one field onto the other that preserves multiplication and addition. Thus we may speak of the finite field with q elements, usually denoted by Fq or GF(q).
74.4.5
Archimedean fields
Main article: Archimedean field An Archimedean field is an ordered field such that for each element there exists a finite expression 1 + 1 + ··· + 1 whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals; or, the field is isomorphic to a subfield of the reals. A necessary condition for an ordered field to be complete is that it be Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence 1/2, 1/3, 1/4, …, every element of which is greater than every infinitesimal, has no limit. (And since every proper subfield of the reals also contains such gaps, up to isomorphism the reals form the unique complete ordered field.)
74.4.6
Field of functions
Given a geometric object X, one can consider functions on such objects. Adding and multiplying them pointwise, i.e., (f ⋅ g)(x) = f(x) ⋅ g(x) this leads to a field. However, for having multiplicative inverses, one has to consider partial functions, which, almost everywhere, are defined and have a non-zero value. If X is an algebraic variety over a field F, then the rational functions X → F form a field, the function field of X. This field consists of the functions that are defined and are the quotient of two polynomial functions outside some subvariety. Likewise, if X is a Riemann surface, then the meromorphic functions S → C form a field. Under certain circumstances, namely when S is compact, S can be reconstructed from this field.
74.4.7
Local and global fields
Another important distinction in the realm of fields, especially with regard to number theory, are local fields and global fields. Local fields are completions of global fields at a given place. For example, Q is a global field, and the attached local fields are Qp and R (Ostrowski’s theorem). Algebraic number fields and function fields over Fq are further global fields. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally—this technique is called local-global principle.
74.5 Some first theorems • Every finite subgroup of the multiplicative group F × is cyclic. This applies in particular to Fq× , it is cyclic of order q − 1. In the introductory example, a generator of F4 × is the element A. • A integral domain is a field if and only if it has no ideals except {0} and itself. Equivalently, an integral domain is a field if and only if its Krull dimension is 0. • Isomorphism extension theorem
74.6. CONSTRUCTING FIELDS
247
74.6 Constructing fields 74.6.1
Closure operations
Assuming the axiom of choice, for every field F, there exists a field F, called the algebraic closure of F, which contains F, is algebraic over F, which means that any element x of F satisfies a polynomial equation fnxn + fn₋₁xn−1 + ··· + f 1 x + f 0 = 0, with coefficients fn, ..., f 0 ∈ F, and is algebraically closed, i.e., any such polynomial does have at least one solution in F. The algebraic closure is unique up to isomorphism inducing the identity on F. However, in many circumstances in mathematics, it is not appropriate to treat F as being uniquely determined by F, since the isomorphism above is not itself unique. In these cases, one refers to such a F as an algebraic closure of F. A similar concept is the separable closure, containing all roots of separable polynomials, instead of all polynomials. For example, if F = Q, the algebraic closure Q is also called field of algebraic numbers. The field of algebraic numbers is an example of an algebraically closed field of characteristic zero; as such it satisfies the same first-order sentences as the field of complex numbers C. In general, all algebraic closures of a field are isomorphic. However, there is in general no preferable isomorphism between two closures. Likewise for separable closures.
74.6.2
Subfields and field extensions
A subfield is, informally, a small field contained in a bigger one. Formally, a subfield E of a field F is a subset containing 0 and 1, closed under the operations +, −, · and multiplicative inverses and with its own operations defined by restriction. For example, the real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers and the rational numbers are examples. The notion of field extension lies at the heart of field theory, and is crucial to many other algebraic domains. A field extension F / E is simply a field F and a subfield E ⊂ F. Constructing such a field extension F / E can be done by “adding new elements” or adjoining elements to the field E. For example, given a field E, the set F = E(X) of rational functions, i.e., equivalence classes of expressions of the kind
p(X) , q(X) where p(X) and q(X) are polynomials with coefficients in E, and q is not the zero polynomial, forms a field. This is the simplest example of a transcendental extension of E. It also is an example of a domain (the ring of polynomials E in this case) being embedded into its field of fractions E(X) . The ring of formal power series E[[X]] is also a domain, and again the (equivalence classes of) fractions of the form p(X)/ q(X) where p and q are elements of E[[X]] form the field of fractions for E[[X]] . This field is actually the ring of Laurent series over the field E, denoted E((X)) . In the above two cases, the added symbol X and its powers did not interact with elements of E. It is possible however that the adjoined symbol may interact with E. This idea will be illustrated by adjoining an element to the field of real numbers R. As explained above, C is an extension of R. C can be obtained from R by adjoining the imaginary symbol i which satisfies i2 = −1. The result is that R[i]=C. This is different from adjoining the symbol X to R, because in that case, the powers of X are all distinct objects, but here, i2 =−1 is actually an element of R. Another way to view this last example is to note that i is a zero of the polynomial p(X) = X2 + 1. The quotient ring R[X]/(X 2 + 1) can be mapped onto C using the map a + bX → a + ib . Since the ideal (X2 +1) is generated by a polynomial irreducible over R, the ideal is maximal, hence the quotient ring is a field. This nonzero ring map from the quotient to C is necessarily an isomorphism of rings. The above construction generalises to any irreducible polynomial in the polynomial ring E[X], i.e., a polynomial p(X) that cannot be written as a product of non-constant polynomials. The quotient ring F = E[X] / (p(X)), is again a field. Alternatively, constructing such field extensions can also be done, if a bigger container is already given. Suppose given a field E, and a field G containing E as a subfield, for example G could be the algebraic closure of E. Let x be
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an element of G not in E. Then there is a smallest subfield of G containing E and x, denoted F = E(x) and called field extension F / E generated by x in G.[10] Such extensions are also called simple extensions. Many extensions are of this type; see the primitive element theorem. For instance, Q(i) is the subfield of C consisting of all numbers of the form a + bi where both a and b are rational numbers. One distinguishes between extensions having various qualities. For example, an extension K of a field k is called algebraic, if every element of K is a root of some polynomial with coefficients in k. Otherwise, the extension is called transcendental. The aim of Galois theory is the study of algebraic extensions of a field.
74.6.3
Rings vs fields
Adding multiplicative inverses to an integral domain R yields the field of fractions of R. For example, the field of fractions of the integers Z is just Q. Also, the field F(X) is the quotient field of the ring of polynomials F[X]. Another method to obtain a field from a commutative ring R is taking the quotient R / m, where m is any maximal ideal of R. The above construction of F = E[X] / (p(X)), is an example, because the irreducibility of the polynomial p(X) is equivalent to the maximality of the ideal generated by this polynomial. Another example are the finite fields Fp = Z / pZ.
74.6.4
Ultraproducts
If I is an index set, U is an ultrafilter on I, and Fi is a field for every i in I, the ultraproduct of the Fi with respect to U is a field. For example, a non-principal ultraproduct of finite fields is a pseudo finite field; i.e., a PAC field having exactly one extension of any degree.
74.7 Galois theory Main article: Galois theory Galois theory aims to study the algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication. The fundamental theorem of Galois theory shows that there is a strong relation between the structure of the symmetry group and the set of algebraic extensions. In the case where F / E is a finite (Galois) extension, Galois theory studies the algebraic extensions of E that are subfields of F. Such fields are called intermediate extensions. Specifically, the Galois group of F over E, denoted Gal(F/E), is the group of field automorphisms of F that are trivial on E (i.e., the bijections σ : F → F that preserve addition and multiplication and that send elements of E to themselves), and the fundamental theorem of Galois theory states that there is a one-to-one correspondence between subgroups of Gal(F/E) and the set of intermediate extensions of the extension F/E. The theorem, in fact, gives an explicit correspondence and further properties. To study all (separable) algebraic extensions of E at once, one must consider the absolute Galois group of E, defined as the Galois group of the separable closure, E sep , of E over E (i.e., Gal(E sep /E). It is possible that the degree of this extension is infinite (as in the case of E = Q). It is thus necessary to have a notion of Galois group for an infinite algebraic extension. The Galois group in this case is obtained as a “limit” (specifically an inverse limit) of the Galois groups of the finite Galois extensions of E. In this way, it acquires a topology.[note 2] The fundamental theorem of Galois theory can be generalized to the case of infinite Galois extensions by taking into consideration the topology of the Galois group, and in the case of E sep /E it states that there this a one-to-one correspondence between closed subgroups of Gal(E sep /E) and the set of all separable algebraic extensions of E (technically, one only obtains those separable algebraic extensions of E that occur as subfields of the chosen separable closure E sep , but since all separable closures of E are isomorphic, choosing a different separable closure would give the same Galois group and thus an “equivalent” set of algebraic extensions).
74.8. GENERALIZATIONS
249
74.8 Generalizations There are also proper classes with field structure, which are sometimes called Fields, with a capital F: • The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. n
• The nimbers form a Field. The set of nimbers with birthday smaller than 22 , the nimbers with birthday smaller than any infinite cardinal are all examples of fields. In a different direction, differential fields are fields equipped with a derivation. For example, the field R(X), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory. Exponential fields, meanwhile, are fields equipped with an exponential function that provides a homomorphism between the additive and multiplicative groups within the field. The usual exponential function makes the real and complex numbers exponential fields, denoted Rₑₓ and Cₑₓ respectively. Generalizing in a more categorical direction yields the field with one element and related objects.
74.8.1
Exponentiation
One does not in general study generalizations of fields with three binary operations. The familiar addition/subtraction, multiplication/division, exponentiation/root-extraction/logarithm operations from the natural numbers to the reals, each built up in terms of iteration of the last, mean that generalizing exponentiation as a binary operation is tempting, but has generally not proven fruitful; instead, an exponential field assumes a unary exponential function from the additive group to the multiplicative group, not a partially defined binary function. Note that the exponential operation of ab is neither associative nor commutative, nor has a unique inverse ( ±2 are both square roots of 4, for instance), √ unlike addition and multiplication, and further is not defined for many pairs—for example, (−1)1/2 = −1 does not define a single number. These all show that even for rational numbers exponentiation is not nearly as well-behaved as addition and multiplication, which is why one does not in general axiomatize exponentiation.
74.9 Applications The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field. Finite fields are used in number theory, Galois theory, cryptography, coding theory and combinatorics; and again the notion of algebraic extension is an important tool.
74.10 See also • Category of fields • Glossary of field theory for more definitions in field theory. • Heyting field • Lefschetz principle • Puiseux series • Ring • Vector space • Vector spaces without fields
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74.11 Notes [1] That is, the axiom for addition only assumes a binary operation + : F × F → F, a, b 7→ a + b. The axiom of inverse allows one to define a unary operation − : F → F a 7→ −a that sends an element to its negative (its additive inverse); this is not taken as given, but is implicitly defined in terms of addition as " −a is the unique b such that a + b = 0 ", “implicitly” because it is defined in terms of solving an equation—and one then defines the binary operation of subtraction, also denoted by "−", as − : F × F → F, a, b 7→ a − b := a + (−b) in terms of addition and additive inverse. In the same way, one defines the binary operation of division ÷ in terms of the assumed binary operation of multiplication and the implicitly defined operation of “reciprocal” (multiplicative inverse). [2] As an inverse limit of finite discrete groups, it is equipped with the profinite topology, making it a profinite topological group
74.12 References [1] Wallace, D A R (1998) Groups, Rings, and Fields, SUMS. Springer-Verlag: 151, Th. 2. [2] Karl Georg Christian v. Staudt, Beiträge zur Geometrie der Lage (Contributions to the Geometry of Position), volume 2 (Nürnberg, (Germany): Bauer and Raspe, 1857). See: “Summen von Würfen” (sums of throws), pp. 166-171 ; “Produckte aus Würfen” (products of throws), pp. 171-176 ; “Potenzen von Würfen” (powers of throws), pp. 176-182. [3] Peter Gustav Lejeune Dirichlet with R. Dedekind, Vorlesungen über Zahlentheorie von P. G. Lejeune Dirichlet (Lectures on Number Theory by P.G. Lejeune Dirichlet), 2nd ed., volume 1 (Braunschweig, Germany: Friedrich Vieweg und Sohn, 1871), p. 424. From page 424: “Unter einem Körper wollen wir jedes System von unendlich vielen reellen oder complexen Zahlen verstehen, welches in sich so abgeschlossen und vollständig ist, dass die Addition, Subtraction, Multiplication und Division von je zwei dieser Zahlen immer wieder eine Zahl desselben Systems hervorbringt.” (By a “field” we will understand any system of infinitely many real or complex numbers, which is so closed and complete that the addition, subtraction, multiplication, and division of any two of these numbers always again produces a number of the same system.) [4] J J O'Connor and E F Robertson, The development of Ring Theory, September 2004. [5] Moore, E. Hastings (1893), “A doubly-infinite system of simple groups”, Bulletin of the New York Mathematical Society 3 (3): 73–78, doi:10.1090/S0002-9904-1893-00178-X, JFM 25.0198.01. From page 75: “Such a system of s marks [i.e., a finite field with s elements] we call a field of order s.” [6] Earliest Known Uses of Some of the Words of Mathematics (F) [7] Fricke, Robert; Weber, Heinrich Martin (1924), Lehrbuch der Algebra, Vieweg, JFM 50.0042.03 [8] Steinitz, Ernst (1910), “Algebraische Theorie der Körper”, Journal für die reine und angewandte Mathematik 137: 167– 309, doi:10.1515/crll.1910.137.167, ISSN 0075-4102, JFM 41.0445.03 [9] Jacobson (2009), p. 213 [10] Jacobson (2009), p. 213
74.13 Sources • Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-13-004763-2, especially Chapter 13 • Allenby, R.B.J.T. (1991), Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2 • Blyth, T.S.; Robertson, E. F. (1985), Groups, rings and fields: Algebra through practice, Cambridge University Press. See especially Book 3 (ISBN 0-521-27288-2) and Book 6 (ISBN 0-521-27291-2). • Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1 • James Ax (1968), The elementary theory of finite fields, Ann. of Math. (2), 88, 239–271
74.14. EXTERNAL LINKS
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74.14 External links • Hazewinkel, Michiel, ed. (2001), “Field”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Field Theory Q&A • Fields at ProvenMath definition and basic properties. • Field at PlanetMath.org.
Chapter 75
Field extension In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field that contains the base field and satisfies additional properties. For instance, the set Q(√2) = {a + b√2 | a, b ∈ Q} is the smallest extension of Q that includes every real solution to the equation x2 = 2.
75.1 Definitions Let L be a field. A subfield of L is a subset K of L that is closed under the field operations of L and under taking inverses in L. In other words, K is a field with respect to the field operations inherited from L. The larger field L is then said to be an extension field of K. To simplify notation and terminology, one says that L / K (read as "L over K") is a field extension to signify that L is an extension field of K. If L is an extension of F which is in turn an extension of K, then F is said to be an intermediate field (or intermediate extension or subextension) of the field extension L /K. Given a field extension L /K and a subset S of L, the smallest subfield of L which contains K and S is denoted by K(S)—i.e. K(S) is the field generated by adjoining the elements of S to K. If S consists of only one element s, K(s) is a shorthand for K({s}). A field extension of the form L = K(s) is called a simple extension and s is called a primitive element of the extension. Given a field extension L /K, the larger field L can be considered as a vector space over K. The elements of L are the “vectors” and the elements of K are the “scalars”, with vector addition and scalar multiplication obtained from the corresponding field operations. The dimension of this vector space is called the degree of the extension and is denoted by [L : K]. An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. Depending on whether the degree is finite or infinite the extension is called a finite extension or infinite extension.
75.2 Caveats The notation L /K is purely formal and does not imply the formation of a quotient ring or quotient group or any other kind of division. Instead the slash expresses the word “over”. In some literature the notation L:K is used. It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an injective ring homomorphism between two fields. Every non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields. Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields. 252
75.3. EXAMPLES
253
75.3 Examples The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extension field of the field of rational numbers Q. Clearly then, C/Q is also a field extension. We have [C : R] = 2 because {1,i} is a basis, so the extension C/R is finite. This is a simple extension because C=R( i ). [R : Q] = c (the cardinality of the continuum), so this extension is infinite. The set Q(√2) = {a + b√2 | a, b ∈ Q} is an extension field of Q, also clearly a simple extension. The degree is 2 because {1, √2} can serve as a basis. Q(√2, √3) = Q(√2)( √3)={a + b√3 | a, b ∈ Q(√2)}={a + b√2+ c√3+ d√6 | a, b,c,d ∈ Q} is an extension field of both Q(√2) and Q, of degree 2 and 4 respectively. Finite extensions of Q are also called algebraic number fields and are important in number theory. Another extension field of the rationals, quite different in flavor, is the field of p-adic numbers Qp for a prime number p. It is common to construct an extension field of a given field K as a quotient ring of the polynomial ring K[X] in order to “create” a root for a given polynomial f(X). Suppose for instance that K does not contain any element x with x2 = −1. Then the polynomial X2 + 1 is irreducible in K[X], consequently the ideal (X2 + 1) generated by this polynomial is maximal, and L = K[X]/(X2 + 1) is an extension field of K which does contain an element whose square is −1 (namely the residue class of X). By iterating the above construction, one can construct a splitting field of any polynomial from K[X]. This is an extension field L of K in which the given polynomial splits into a product of linear factors. If p is any prime number and n is a positive integer, we have a finite field GF(pn ) with pn elements; this is an extension field of the finite field GF(p) = Z/pZ with p elements. Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; the elements of K(X) are fractions of two polynomials over K, and indeed K(X) is the field of fractions of the polynomial ring K[X]. This field of rational functions is an extension field of K. This extension is infinite. Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by C(M). It is an extension field of C, if we identify every complex number with the corresponding constant function defined on M. Given an algebraic variety V over some field K, then the function field of V, consisting of the rational functions defined on V and denoted by K(V), is an extension field of K.
75.4 Elementary properties If L/K is a field extension, then L and K share the same 0 and the same 1. The additive group (K,+) is a subgroup of (L,+), and the multiplicative group (K−{0},·) is a subgroup of (L−{0},·). In particular, if x is an element of K, then its additive inverse −x computed in K is the same as the additive inverse of x computed in L; the same is true for multiplicative inverses of non-zero elements of K. In particular then, the characteristics of L and K are the same.
75.5 Algebraic and transcendental elements and extensions If L is an extension of K, then an element of L which is a root of a nonzero polynomial over K is said to be algebraic over K. Elements that are not algebraic are called transcendental. For example: • In C/R, i is algebraic because it is a root of x2 + 1. • In R/Q, √2 + √3 is algebraic, because it is a root[1] of x4 − 10x2 + 1 • In R/Q, e is transcendental because there is no polynomial with rational coefficients that has e as a root (see transcendental number) • In C/R, e is algebraic because it is the root of x − e The special case of C/Q is especially important, and the names algebraic number and transcendental number are used to describe the complex numbers that are algebraic and transcendental (respectively) over Q.
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If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension; otherwise it is said to be a transcendental extension. A subset S of L is called algebraically independent over K if no non-trivial polynomial relation with coefficients in K exists among the elements of S. The largest cardinality of an algebraically independent set is called the transcendence degree of L/K. It is always possible to find a set S, algebraically independent over K, such that L/K(S) is algebraic. Such a set S is called a transcendence basis of L/K. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension L/K is said to be purely transcendental if and only if there exists a transcendence basis S of L/K such that L=K(S). Such an extension has the property that all elements of L except those of K are transcendental over K, but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the form L/K where both L and K are algebraically closed. In addition, if L/K is purely transcendental and S is a transcendence basis of the extension, it doesn't necessarily follow that L=K(S). (For example, consider the extension Q(x,√x)/Q, where x is transcendental over Q. The set {x} is algebraically independent since x is transcendental. Obviously, the extension Q(x,√x)/Q(x) is algebraic, hence {x} is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in x for √x. But it is easy to see that {√x} is a transcendence basis that generates Q(x,√x)), so this extension is indeed purely transcendental.) It can be shown that an extension is algebraic if and only if it is the union of its finite subextensions. In particular, every finite extension is algebraic. For example, • C/R and Q(√2)/Q, being finite, are algebraic. • R/Q is transcendental, although not purely transcendental. • K(X)/K is purely transcendental. A simple extension is finite if generated by an algebraic element, and purely transcendental if generated by a transcendental element. So • R/Q is not simple, as it is neither finite nor purely transcendental. Every field K has an algebraic closure; this is essentially the largest extension field of K which is algebraic over K and which contains all roots of all polynomial equations with coefficients in K. For example, C is the algebraic closure of R.
75.6 Normal, separable and Galois extensions An algebraic extension L/K is called normal if every irreducible polynomial in K[X] that has a root in L completely factors into linear factors over L. Every algebraic extension F/K admits a normal closure L, which is an extension field of F such that L/K is normal and which is minimal with this property. An algebraic extension L/K is called separable if the minimal polynomial of every element of L over K is separable, i.e., has no repeated roots in an algebraic closure over K. A Galois extension is a field extension that is both normal and separable. A consequence of the primitive element theorem states that every finite separable extension has a primitive element (i.e. is simple). Given any field extension L/K, we can consider its automorphism group Aut(L/K), consisting of all field automorphisms α: L → L with α(x) = x for all x in K. When the extension is Galois this automorphism group is called the Galois group of the extension. Extensions whose Galois group is abelian are called abelian extensions. For a given field extension L/K, one is often interested in the intermediate fields F (subfields of L that contain K). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a bijection between the intermediate fields and the subgroups of the Galois group, described by the fundamental theorem of Galois theory.
75.7. GENERALIZATIONS
255
75.7 Generalizations Field extensions can be generalized to ring extensions which consist of a ring and one of its subrings. A closer noncommutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. CSAs can be further generalized to Azumaya algebras, where the base field is replaced by a commutative local ring.
75.8 Extension of scalars Main article: Extension of scalars Given a field extension, one can "extend scalars" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via complexification. In addition to vector spaces, one can perform extension of scalars for associative algebras defined over the field, such as polynomials or group algebras and the associated group representations. Extension of scalars of polynomials is often used implicitly, by just considering the coefficients as being elements of a larger field, but may also be considered more formally. Extension of scalars has numerous applications, as discussed in extension of scalars: applications.
75.9 See also • Field theory • Glossary of field theory • Tower of fields • Primary extension • Regular extension
75.10 Notes [1] “Wolfram|Alpha input: sqrt(2)+sqrt(3)". Retrieved 2010-06-14.
75.11 References • Lang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4
75.12 External links • Hazewinkel, Michiel, ed. (2001), “Extension of a field”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4
Chapter 76
Field norm In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
76.1 Formal definition Let K be a field and L a finite extension (and hence an algebraic extension) of K. The field L is then a finite dimensional vector space over K. Multiplication by α, an element of L,
mα : L → L by given mα (x) = αx is a K-linear transformation of this vector space into itself. The norm, NL/K(α), is defined as the determinant of this linear transformation.[1] For nonzero α in L, let σ1 (α), ..., σ (α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of L), then NL/K (α) =
n ∏
[L:K(α)] σj (α)
j=1
If L/K is separable then each root appears only once in the product (the exponent [L:K(α)] may still be greater than 1). More particularly, if L/K is a Galois extension and α is in L, then the norm of α is the product of all the Galois conjugates of α, i.e.
NL/K (α) =
∏
g(α)
g∈Gal(L/K)
where Gal(L/K) denotes the Galois group of L/K.[2]
76.2 Example The field norm from the complex numbers to the real numbers sends x + iy to 256
76.3. PROPERTIES OF THE NORM
257
x2 + y2 , because the Galois group of C over R has two elements, the identity element and complex conjugation, and taking the product yields (x + iy)(x − iy) = x2 + y2 . In this example the norm was the square of the usual Euclidean distance norm in C . In general, the field norm is very different from the usual distance norm. √ We will illustrate that with an example where the field norm can be negative. Consider the number√ field K √ = Q( 2) . The Galois group of K over Q has order d = 2 and is generated by the √ element which sends 2 to − 2 . So the norm of 1 + 2 is:
(1 +
√ √ 2)(1 − 2) = −1.
√ √ The field norm can also be obtained Q -basis of Q( 2) , say {1, 2} : then √ without the Galois √ group. √ Fix a √ multiplication by the number 1 + 2 sends 1 to 1 + 2 and 2 to 2 + 2 . So the determinant of “multiplying by √ 1 + 2 is the determinant of the matrix which sends the vector (1, 0)T (corresponding to the first basis element, i.e. √ 1) to (1, 1)T and the vector (0, 1)T (which represents the second basis element 2 ) to (2, 1)T , viz.: [
] 1 2 . 1 1
The determinant of this matrix is −1.
76.3 Properties of the norm Several properties of the norm function hold for any finite extension.[3] The norm NL/K : L* → K* is a group homomorphism from the multiplicative group of L to the multiplicative group of K, that is NL/K (αβ) = NL/K (α) NL/K (β) all for α, β ∈ L∗ Furthermore, if a in K: NL/K (aα) = a[L:K] NL/K (α) all for α ∈ L If a ∈ K then NL/K (a) = a[L:K] . Additionally, norm behaves well in towers of fields: if M is a finite extension of L, then the norm from M to K is just the composition of the norm from M to L with the norm from L to K, i.e. NM /K = NL/K ◦ NM /L
76.4 Finite fields Let L = GF(qn ) be a finite extension of a finite field K = GF(q). Since L/K is a Galois extension, if α is in L, then the norm of α is the product of all the Galois conjugates of α, i.e.[4] NL/K (α) = α • αq • · · · • αq
n−1
= α(q
n
−1)/(q−1)
In this setting we have the additional properties,[5] • NL/K (αq ) = NL/K (α) all for α ∈ L • any fora ∈ K, have we NL/K (a) = an .
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CHAPTER 76. FIELD NORM
76.5 Further properties The norm of an algebraic integer is again an integer, because it is equal (up to sign) to the constant term of the characteristic polynomial. In algebraic number theory one defines also norms for ideals. This is done in such a way that if I is an ideal of OK, the ring of integers of the number field K, N(I) is the number of residue classes in OK /I – i.e. the cardinality of this finite ring. Hence this norm of an ideal is always a positive integer. When I is a principal ideal αOK then N(I) is equal to the absolute value of the norm to Q of α, for α an algebraic integer.
76.6 See also • Field trace • Ideal norm
76.7 Notes [1] Rotman 2002, p. 940 [2] Rotman 2002, p. 943 [3] Roman 1995, p. 151 (1st ed.) [4] Lidl & Niederreiter 1997, p.57 [5] Mullen & Panario 2013, p. 21
76.8 References • Lidl, Rudolf; Niederreiter, Harald (1997) [1983], Finite Fields, Encyclopedia of Mathematics and its Applications 20 (Second ed.), Cambridge University Press, ISBN 0-521-39231-4, Zbl 0866.11069 • Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6 • Roman, Steven (2006), Field theory, Graduate Texts in Mathematics 158 (Second ed.), Springer, Chapter 8, ISBN 978-0-387-27677-9, Zbl 1172.12001 • Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 978-0-13-087868-7
Chapter 77
Field of fractions “Quotient field” redirects here. It should not be confused with a quotient ring. In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R are equivalence classes (see the construction below) written as ab with a and b in R and b ̸= 0 . The field of fractions of R is sometimes denoted by Quot(R) or Frac(R) . Mathematicians refer to this construction as the field of fractions, fraction field, field of quotients, or quotient field. All four are in common usage. The expression “quotient field” may sometimes run the risk of confusion with the quotient of a ring by an ideal, which is a quite different concept.
77.1 Examples • The field of fractions of the ring of integers is the field of rationals, Q = Quot(Z) . • Let R := {a + bi|a, b ∈ Z} be the ring of Gaussian integers. Then Quot(R) = {c + di|c, d ∈ Q} , the field of Gaussian rationals. • The field of fractions of a field is canonically isomorphic to the field itself. • Given a field K , the field of fractions of the polynomial ring in one indeterminate K[X] (which is an integral domain), is called the field of rational functions or field of rational fractions[1][2][3] and is denoted K(X) .
77.2 Construction Let R be any integral domain. For n, d ∈ R with d ̸= 0 , the fraction nd denotes the equivalence class of pairs (n, d) , where (n, d) is equivalent to (m, b) if and only if nb = md . (The definition of equivalence is modelled on the property of rational numbers that nd = m b if and only if nb = md .) The field of fractions Quot(R) is defined as the nb+md nm set of all such fractions nd . The sum of nd and m , and the product of nd and m b is defined as db b is defined as db (one checks that these are well defined). The embedding of R in Quot(R) maps each n in R to the fraction en e for any nonzero e ∈ R (the equivalence class is independent of the choice e ). This is modelled on the identity n1 = n . If additionally, R contains a multiplicative n identity (that is, R is an integral domain), then en e = 1 . The field of fractions of R is characterised by the following universal property: if h : R → F is an injective ring homomorphism from R into a field F , then there exists a unique ring homomorphism g : Quot(R) → F which extends h . There is a categorical interpretation of this construction. Let C be the category of integral domains and injective ring maps. The functor from C to the category of fields which takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the left adjoint of the forgetful functor from the category of fields to C . 259
260
CHAPTER 77. FIELD OF FRACTIONS
A multiplicative identity is not required for the role of the integral domain; this construction can be applied to any nonzero commutative rng with no nonzero zero divisors.[4]
77.3 Generalisation Main article: Localization of a ring For any commutative ring R and any multiplicative set S in R , the localization S −1 R is the commutative ring consisting of fractions rs with r ∈ R and s ∈ S , where now (r, s) is equivalent to (r′ , s′ ) if and only if there exists t ∈ S such that t(rs′ − r′ s) = 0 . Two special cases of this are notable: • If S is the complement of a prime ideal P , then S −1 R is also denoted RP . When R is an integral domain and P is the zero ideal, RP is the field of fractions of R . • If S is the set of non-zero-divisors in R , then S −1 R is called the total quotient ring. The total quotient ring of an integral domain is its field of fractions, but the total quotient ring is defined for any commutative ring.
77.4 See also • Ore condition; this is the condition one needs to consider in the noncommutative case. • Projective line over a ring; alternative structure not limited to integral domains.
77.5 References [1] Ėrnest Borisovich Vinberg (2003). A course in algebra. p. 131. [2] Stephan Foldes (1994). Fundamental structures of algebra and discrete mathematics. p. 128. [3] Pierre Antoine Grillet (2007). Abstract algebra. p. 124. [4] Rings, Modules, and Linear Algebra: Hartley, B & Hawkes, T.O. 1970
Chapter 78
Field trace For other uses, see Trace In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L to K.
78.1 Definition Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L, mα : L → L by given mα (x) = αx is a K-linear transformation of this vector space into itself. The trace, TrL/K(α), is defined as the (linear algebra) trace of this linear transformation.[1] For α in L, let σ1 (α), ..., σ (α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of L), then
TrL/K (α) = [L : K(α)]
n ∑
σj (α)
j=1
If L/K is separable then each root appears only once and the coefficient above is one.[2] More particularly, if L/K is a Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.
TrL/K (α) =
∑
g(α)
g∈Gal(L/K)
where Gal(L/K) denotes the Galois group of L/K.
78.2 Example √ √ √ Let L = Q( d) be a quadratic extension of Q . Then a basis of L/Q is {1, d}. If α = a + b d then the matrix of mα is: [
a bd b a
]
261
262
CHAPTER 78. FIELD TRACE
and so, TrL/Q (α) = 2a .[1] The minimal polynomial of α is X2 - 2a X + a2 - d b2 .
78.3 Properties of the trace Several properties of the trace function hold for any finite extension.[3] The trace TrL/K : L → K is a K-linear map (a K-linear functional), that is
TrL/K (αa + βb) = α TrL/K (a) + β TrL/K (b) all for α, β ∈ K If α ∈ K then TrL/K (α) = [L : K]α. Additionally, trace behaves well in towers of fields: if M is a finite extension of L, then the trace from M to K is just the composition of the trace from M to L with the trace from L to K, i.e.
TrM /K = TrL/K ◦ TrM /L
78.4 Finite fields Let L = GF(qn ) be a finite extension of a finite field K = GF(q). Since L/K is a Galois extension, if α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.[4]
TrL/K (α) = α + αq + · · · + αq
n−1
In this setting we have the additional properties,[5] • TrL/K (aq ) = TrL/K (a) for a ∈ L • any forα ∈ K, have we |{b ∈ L : TrL/K (b) = α}| = q n−1 And,[6] Theorem. For b ∈ L, let Fb be the map a 7→ TrL/K (ba). Then Fb ≠ Fc if b ≠ c. Moreover the K-linear transformations from L to K are exactly the maps of the form Fb as b varies over the field L. When K is the prime subfield of L, the trace is called the absolute trace and otherwise it is a relative trace.[4]
78.4.1
Application
A quadratic equation, ax2 + bx + c = 0, with a ̸= 0, and coefficients in the finite field GF(q) = Fq has either 0, 1 or 2 roots in GF(q) (and two roots, counted with multiplicity, in the quadratic extension GF(q2 )). If the characteristic of GF(q) is odd, the discriminant, Δ = b2 - 4ac indicates the number of roots in GF(q) and the classical quadratic formula gives the roots. However, when GF(q) has even characteristic (i.e., q = 2h for some positive integer h), these formulas are no longer applicable. Consider the quadratic equation ax2 +√bx + c = 0 with coefficients in the finite field GF(2h ).[7] If b = 0 then this equation has the unique solution x = ac in GF(q). If b ≠ 0 then the substitution y = ax/b converts the quadratic equation to the form:
y 2 + y + δ = 0, where δ =
ac b2
This equation has two solutions in GF(q) if and only if the absolute trace TrGF (q)/GF (2) (δ) = 0. In this case, if y = s is one of the solutions, then y = s + 1 is the other. Let k be any element of GF(q) with TrGF (q)/GF (2) (k) = 1. Then a solution to the equation is given by:
78.5. TRACE FORM
263
h−2
y = s = kδ 2 + (k + k 2 )δ 4 + . . . + (k + k 2 + . . . + k 2
h−1
)δ 2
When h = 2m + 1, a solution is given by the simpler expression:
2
4
2m
y = s = δ + δ2 + δ2 + . . . + δ2
78.5 Trace form When L/K is separable, the trace provides a duality theory via the trace form: the map from L × L to K sending (x, y) to TrL/K(xy) is a nondegenerate, symmetric, bilinear form called the trace form. An example of where this is used is in algebraic number theory in the theory of the different ideal. The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K.[8] The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.[8] If L/K is an inseparable extension, then the trace form is identically 0.[9]
78.6 See also • Field norm • Reduced trace
78.7 Notes [1] Rotman 2002, p. 940 [2] Rotman 2002, p. 941 [3] Roman 1995, p. 151 (1st ed.) [4] Lidl & Niederreiter 1997, p.54 [5] Mullen & Panario 2013, p. 21 [6] Lidl & Niederreiter 1997, p.56 [7] Hirschfeld 1979, pp. 3-4 [8] Lorenz (2008) p.38 [9] Isaacs 1994, p. 369 as footnoted in Rotman 2002, p. 943
78.8 References • Hirschfeld, J.W.P. (1979), Projective Geometries over Finite Fields, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-853526-0 • Isaacs, I.M. (1994), Algebra, A Graduate Course, Brooks/Cole Publishing • Lidl, Rudolf; Niederreiter, Harald (1997) [1983], Finite Fields, Encyclopedia of Mathematics and its Applications 20 (Second ed.), Cambridge University Press, ISBN 0-521-39231-4, Zbl 0866.11069 • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.
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CHAPTER 78. FIELD TRACE
• Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6 • Roman, Steven (2006), Field theory, Graduate Texts in Mathematics 158 (Second ed.), Springer, Chapter 8, ISBN 978-0-387-27677-9, Zbl 1172.12001 • Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 978-0-13-087868-7
78.9 Further reading • Conner, P.E.; Perlis, R. (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017. • Section VI.5 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556
Chapter 79
Finite extensions of local fields In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups. In this article, a local field is non-archimedean and has finite residue field.
79.1 Unramified extension Let L/K be a finite Galois extension of nonarchimedean local fields with finite residue fields l/k and Galois group G . Then the following are equivalent. • (i) L/K is unramified. • (ii) OL /OL p is a field, where p is the maximal ideal of OK . • (iii) [L : K] = [l : k] • (iv) The inertia subgroup of G is trivial. • (v) If π is a uniformizing element of K , then π is also a uniformizing element of L . When L/K is unramified, by (iv) (or (iii)), G can be identified with Gal(l/k) , which is finite cyclic. The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.
79.2 Totally ramified extension Again, let L/K be a finite Galois extension of nonarchimedean local fields with finite residue fields l/k and Galois group G . The following are equivalent. • L/K is totally ramified • G coincides with its inertia subgroup. • L = K[π] where π is a root of an Eisenstein polynomial. • The norm N (L/K) contains a uniformizer of K .
79.3 See also • Abhyankar’s lemma 265
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79.4 References • Cassels, J.W.S. (1986). Local Fields. London Mathematical Society Student Texts 3. Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006. • Weiss, Edwin (1976). Algebraic Number Theory (2nd unaltered ed.). Chelsea Publishing. ISBN 0-8284-02930. Zbl 0348.12101.
Chapter 80
Formal group In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by S. Bochner (1946). The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology.
80.1 Definitions A one-dimensional formal group law over a commutative ring R is a power series F(x,y) with coefficients in R, such that 1. F(x,y) = x + y + terms of higher degree 2. F(x, F(y,z)) = F(F(x,y), z) (associativity). The simplest example is the additive formal group law F(x, y) = x + y. The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin. More generally, an n-dimensional formal group law is a collection of n power series Fi(x1 , x2 , ..., xn, y1 , y2 , ..., yn) in 2n variables, such that 1. F(x,y) = x + y + terms of higher degree 2. F(x, F(y,z)) = F(F(x,y), z) where we write F for (F 1 , ..., Fn), x for (x1 ,..., xn), and so on. The formal group law is called commutative if F(x,y) = F(y,x). Prop. If R is Z -torsion free then any one-dimensional formal group law over R is commutative. Proof. The torsion freeness gives us the exponential and logarithm which allows us to write F as F(x,y) = exp(log(x) + log(y)). There is no need for an axiom analogous to the existence of an inverse for groups, as this turns out to follow automatically from the definition of a formal group law. In other words we can always find a (unique) power series G such that F(x,G(x)) = 0. A homomorphism from a formal group law F of dimension m to a formal group law G of dimension n is a collection f of n power series in m variables, such that G(f(x), f(y)) = f(F(x, y)). 267
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A homomorphism with an inverse is called an isomorphism, and is called a strict isomorphism if in addition f(x)= x + terms of higher degree. Two formal group laws with an isomorphism between them are essentially the same; they differ only by a “change of coordinates”.
80.2 Examples • The additive formal group law is given by
F (x, y) = x + y. • The multiplicative formal group law is given by
F (x, y) = x + y + xy. This rule can be understood as follows. The product G in the (multiplicative group of the) ring R is given by G(a,b) = ab. If we “change coordinates” to make 0 the identity by putting a = 1 + x, b = 1 + y, and G = 1 + F, then we find that F(x, y) = x + y + xy. Over the rational numbers, there is an isomorphism from the additive formal group law to the multiplicative one, given by exp(x) − 1. Over general commutative rings R there is no such homomorphism as defining it requires non-integral rational numbers, and the additive and multiplicative formal groups are usually not isomorphic. • More generally, we can construct a formal group law of dimension n from any algebraic group or Lie group of dimension n, by taking coordinates at the identity and writing down the formal power series expansion of the product map. The additive and multiplicative formal group laws are obtained in this way from the additive and multiplicative algebraic groups. Another important special case of this is the formal group (law) of an elliptic curve (or abelian variety). • F(x,y) = (x + y)/(1 + xy) is a formal group law coming from the addition formula for the hyperbolic tangent function: tanh(x + y) = F(tanh(x), tanh(y)), and is also the formula for addition of velocities in special relativity (with the speed of light equal to 1). √ √ • F (x, y) = (x 1 − y 4 + y 1 − x4 )/(1 + x2 y 2 ) is a formal group law over Z[1/2] found by Euler, in the form of the addition formula for an elliptic integral:
∫ 0
x
dt √ + 1 − t4
∫ 0
y
dt √ = 1 − t4
∫
F (x,y) 0
dt √ . 1 − t4
80.3 Lie algebras Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part F 2 of the formal group law. [x,y] = F 2 (x,y) − F 2 (y,x) The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group laws, followed by taking the Lie algebra of the formal group: Lie groups → Formal group laws → Lie algebras
80.4. THE LOGARITHM OF A COMMUTATIVE FORMAL GROUP LAW
269
Over fields of characteristic 0, formal group laws are essentially the same as finite-dimensional Lie algebras: more precisely, the functor from finite-dimensional formal group laws to finite-dimensional Lie algebras is an equivalence of categories. Over fields of non-zero characteristic, formal group laws are not equivalent to Lie algebras. In fact, in this case it is well known that passing from an algebraic group to its Lie algebra often throws away too much information, but passing instead to the formal group law often keeps enough information. So in some sense formal group laws are the “right” substitute for Lie algebras in characteristic p > 0.
80.4 The logarithm of a commutative formal group law If F is a commutative n-dimensional formal group law over a commutative Q-algebra R, then it is strictly isomorphic to the additive formal group law. In other words, there is a strict isomorphism f from the additive formal group to F, called the logarithm of F, so that f(F(x,y)) = f(x) + f(y) Examples: • The logarithm of F(x, y) = x + y is f(x) = x. • The logarithm of F(x, y) = x + y + xy is f(x) = log(1 + x), because log(1 + x + y + xy) = log(1 + x) + log(1 + y). If R does not contain the rationals, a map f can be constructed by extension of scalars to R⊗Q, but this will send everything to zero if R has positive characteristic. Formal group laws over a ring R are often constructed by writing down their logarithm as a power series with coefficients in R⊗Q, and then proving that the coefficients of the corresponding formal group over R⊗Q actually lie in R. When working in positive characteristic, one typically replaces R with a mixed characteristic ring that has a surjection to R, such as the ring W(R) of Witt vectors, and reduces to R at the end.
80.5 The formal group ring of a formal group law The formal group ring of a formal group law is a cocommutative Hopf algebra analogous to the group ring of a group and to the universal enveloping algebra of a Lie algebra, both of which are also cocommutative Hopf algebras. In general cocommutative Hopf algebras behave very much like groups. For simplicity we describe the 1-dimensional case; the higher-dimensional case is similar except that notation becomes messier. Suppose that F is a (1-dimensional) formal group law over R. Its formal group ring (also called its hyperalgebra or its covariant bialgebra) is a cocommutative Hopf algebra H constructed as follows. • As an R-module, H is free with a basis 1 = D(0) , D(1) , D(2) , ... • The coproduct Δ is given by ΔD(n) = ∑D(i) ⊗ D(n−i) (so the dual of this coalgebra is just the ring of formal power series). • The counit η is given by the coefficient of D(0) . • The identity is 1 = D(0) . • The antipode S takes D(n) to (−1)n D(n) . • The coefficient of D(1) in the product D(i) D(j) is the coefficient of xi yj in F(x, y). Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it. So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure is given above.
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80.6 Formal group laws as functors Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F(S) whose underlying set is N n where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of N n ; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms. This makes F into a functor from commutative R-algebras S to groups. We can extend the definition of F(S) to some topological R-algebras. In particular, if S is an inverse limit of discrete R algebras, we can define F(S) to be the inverse limit of the corresponding groups. For example, this allows us to define F(Zp) with values in the p-adic numbers. The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional; the general case is similar. For any cocommutative Hopf algebra, an element g is called group-like if Δg = g ⊗ g and εg = 1, and the group-like elements form a group under multiplication. In the case of the Hopf algebra of a formal group law over a ring, the group like elements are exactly those of the form D(0) + D(1) x + D(2) x2 + ... for nilpotent elements x. In particular we can identify the group-like elements of H⊗S with the nilpotent elements of S, and the group structure on the group-like elements of H⊗S is then identified with the group structure on F(S).
80.7 The height of a formal group law Suppose that f is a homomorphism between one dimensional formal group laws over a field of characteristic p > 0. h Then f is either zero, or the first nonzero term in its power series expansion is axp for some non-negative integer h, called the height of the homomorphism f. The height of the zero homomorphism is defined to be ∞. The height of a one dimensional formal group law over a field of characteristic p > 0 is defined to be the height of its multiplication by p map. Two one dimensional formal group laws over an algebraically closed field of characteristic p > 0 are isomorphic if and only if they have the same height, and the height can be any positive integer or ∞. Examples: • The additive formal group law F(x, y) = x + y has height ∞, as its pth power map is 0. • The multiplicative formal group law F(x, y) = x + y + xy has height 1, as its pth power map is (1 + x)p − 1 = xp . • The formal group law of an elliptic curve has height either one or two, depending on whether the curve is ordinary or supersingular. Supersingularity can be detected by the vanishing of the Eisenstein series Ep−1 .
80.8 Lazard ring Main article: Lazard’s universal ring There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let F(x, y) be x + y + Σci,j xi yj for indeterminates
80.9. FORMAL GROUPS
271
ci,j, and we define the universal ring R to be the commutative ring generated by the elements ci,j, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring R has the following universal property: For any commutative ring S, one-dimensional formal group laws over S correspond to ring homomorphisms from R to S. The commutative ring R constructed above is known as Lazard’s universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degrees 2, 4, 6, ... (where ci,j has degree 2(i + j − 1)). Daniel Quillen proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard’s universal ring, explaining the unusual grading.
80.9 Formal groups A formal group is a group object in the category of formal schemes. • If G is a functor from Artinian algebras to groups which is left exact, then it representable (G is the functor of points of a formal group. (left exactness of a functor is equivalent to commuting with finite projective limits). b , the formal completion of G at the identity has the structure of a formal group. • If G is a group scheme then G • A smooth group scheme is isomorphic to Spf(R[[T1 , . . . , Tn ]]) . Some people call a formal group scheme smooth if the converse holds. • formal smoothness asserts the existence of lifts of deformations and can apply to formal schemes that are larger than points. A smooth formal group scheme is a special case of a formal group scheme. • Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections. • The (non-strict) isomorphisms between formal group laws induced by change of parameters make up the elements of the group of coordinate changes on the formal group. Formal groups and formal group laws can also be defined over arbitrary schemes, rather than just over commutative rings or fields, and families can be classified by maps from the base to a parametrizing object. The moduli space of formal group laws is a disjoint union of infinite-dimensional affine spaces, whose components are parametrized by dimension, and whose points are parametrized by admissible coefficients of the power series F. The corresponding moduli stack of smooth formal groups is a quotient of this space by a canonical action of the infinite-dimensional groupoid of coordinate changes. Over an algebraically closed field, the substack of one dimensional formal groups is either a point (in characteristic zero) or an infinite chain of stacky points parametrizing heights. In characteristic zero, the closure of each point contains all points of greater height. This difference gives formal groups a rich geometric theory in positive and mixed characteristic, with connections to the Steenrod algebra, p-divisible groups, Dieudonné theory, and Galois representations. For example, the Serre-Tate theorem implies that the deformations of a group scheme are strongly controlled by those of its formal group, especially in the case of supersingular abelian varieties. For supersingular elliptic curves, this control is complete, and this is quite different from the characteristic zero situation where the formal group has no deformations. A formal group is sometimes defined as a cocommutative Hopf algebra (usually with some extra conditions added, such as being pointed or connected).[1] This is more or less dual to the notion above. In the smooth case, choosing coordinates is equivalent to taking a distinguished basis of the formal group ring. Some authors use the term formal group to mean formal group law.
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80.10 Lubin–Tate formal group laws Main article: Lubin–Tate formal group law We let Zp be the ring of p-adic integers. The Lubin–Tate formal group law is the unique (1-dimensional) formal group law F such that e(x) = px + xp is an endomorphism of F, in other words
e(F (x, y)) = F (e(x), e(y)). More generally we can allow e to be any power series such that e(x) = px + higher-degree terms and e(x) = xp mod p. All the group laws for different choices of e satisfying these conditions are strictly isomorphic.[2] For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax + higher-degree terms. This gives an action of the ring Zp on the Lubin–Tate formal group law. There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue class field.[3] This construction was introduced by Lubin & Tate (1965), in a successful effort to isolate the local field part of the classical theory of complex multiplication of elliptic functions. It is also a major ingredient in some approaches to local class field theory.[4]
80.11 See also • Witt vector • Artin–Hasse exponential
80.12 References [1] Underwood, Robert G. (2011). An introduction to Hopf algebras. Berlin: Springer-Verlag. p. 121. ISBN 978-0-38772765-3. Zbl 1234.16022. [2] Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences 49 (Second ed.). p. 168. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002. [3] Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. pp. 62–63. ISBN 3-540-63003-1. Zbl 0819.11044. [4] e.g. Serre, Jean-Pierre (1967). “Local class field theory”. In Cassels, J.W.S.; Fröhlich, Albrecht. Algebraic Number Theory. Academic Press. pp. 128–161. Zbl 0153.07403.Hazewinkel, Michiel (1975). “Local class field theory is easy”. Advances in Math. 18 (2): 148–181. doi:10.1016/0001-8708(75)90156-5. Zbl 0312.12022.Iwasawa, Kenkichi (1986). Local class field theory. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press. ISBN 978-0-19-504030-2. MR 863740. Zbl 0604.12014.
• Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 9780-226-00524-9 • Bochner, Salomon (1946), “Formal Lie groups”, Annals of Mathematics. Second Series 47: 192–201, ISSN 0003-486X, JSTOR 1969242, MR 0015397 • M. Demazure, Lectures on p-divisible groups Lecture Notes in Mathematics, 1972. ISBN 0-387-06092-8 • Fröhlich, A. (1968), Formal groups, Lecture Notes in Mathematics 74, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0074373, MR 0242837 • P. Gabriel, Étude infinitésimale des schémas en groupes SGA 3 Exp. VIIB • Formal Groups and Applications (Pure and Applied Math 78) Michiel Hazewinkel Publisher: Academic Pr (June 1978) ISBN 0-12-335150-2
80.12. REFERENCES
273
• Lazard, Michel (1975), Commutative formal groups, Lecture Notes in Mathematics 443, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070554, ISBN 978-3-540-07145-7, MR 0393050 • Lubin, Jonathan; Tate, John (1965), “Formal complex multiplication in local fields”, Annals of Mathematics. Second Series 81: 380–387, ISSN 0003-486X, JSTOR 1970622, MR 0172878, Zbl 0128.26501 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Strickland, N. “Formal groups” (PDF). • Zarkhin, Yu.G. (2001), “F/f040820”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 81
Formally real field In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be expanded with a (not necessarily unique) ordering that makes it an ordered field.
81.1 Alternative Definitions The definition given above is not a first-order definition, as it requires quantifiers over sets. However, the following criteria can be coded as first-order sentences in the language of fields, and are equivalent to the above definition. A formally real field F is a field that satisfies in addition one of the following equivalent properties:[1][2] • −1 is not a sum of squares in F. In other words, the Stufe of F is infinite. (In particular, such a field must have characteristic 0, since in a field of characteristic p the element −1 is a sum of 1’s.) • There exists an element of F that is not a sum of squares in F, and the characteristic of F is not 2. • If any sum of squares of elements of F equals zero, then each of those elements must be zero. It is easy to see that these three properties are equivalent. It is also easy to see that a field that admits an ordering must satisfy these three properties. A proof that if F satisfies these three properties, then F admits an ordering uses the notion of prepositive cones and positive cones. Suppose −1 is not a sum of squares, then a Zorn’s Lemma argument shows that the prepositive cone of sums of squares can be extended to a positive cone P⊂F. One uses this positive cone to define an ordering: a≤b if and only if b-a belongs to P.
81.2 Real Closed Fields A formally real field with no formally real proper algebraic extension is a real closed field.[3] If K is formally real and Ω is an algebraically closed field containing K, then there is a real closed subfield of Ω containing K. A real closed field can be ordered in a unique way.[3]
81.3 Notes [1] Rajwade, Theorem 15.1. [2] Milnor and Husemoller (1973) p.60 [3] Rajwade (1993) p.216
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81.4 References • Milnor, John; Husemoller, Dale (1973). Symmetric bilinear forms. Springer. ISBN 3-540-06009-X. • Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
Chapter 82
Fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.
82.1 Definition and basic results Let R be an integral domain, and let K be its field of fractions. A fractional ideal of R is an R-submodule I of K such that there exists a non-zero r ∈ R such that rI ⊆ R. The element r can be thought of as clearing out the denominators in I. The principal fractional ideals are those R-submodules of K generated by a single nonzero element of K. A fractional ideal I is contained in R if, and only if, it is an ('integral') ideal of R. A fractional ideal I is called invertible if there is another fractional ideal J such that IJ = R (where IJ = { a1 b1 + a2 b2 + ... + anbn : ai ∈ I, bi ∈ J, n ∈ Z>₀ } is called the product of the two fractional ideals). In this case, the fractional ideal J is uniquely determined and equal to the generalized ideal quotient
(R : I) = {x ∈ K : xI ⊆ R}. The set of invertible fractional ideals form an abelian group with respect to above product, where the identity is the unit ideal R itself. This group is called the group of fractional ideals of R. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an R-module. Every finitely generated R-submodule of K is a fractional ideal and if R is noetherian these are all the fractional ideals of R.
82.2 Dedekind domains In Dedekind domains, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. In fact, this property characterizes Dedekind domains: an integral domain is a Dedekind domain if, and only if, every non-zero fractional ideal is invertible. The quotient group of fractional ideals by the subgroup of principal fractional ideals is an important invariant of a Dedekind domain called the ideal class group.
82.3 Divisorial ideal Let I˜ denote the intersection of all principal fractional ideals containing a nonzero fractional ideal I. Equivalently, 276
82.4. SEE ALSO
277
I˜ = (R : (R : I)), where as above
(R : I) = {x ∈ K : xI ⊆ R}. If I˜ = I then I is called divisorial.[1] In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals. If I is divisorial and J is a nonzero fractional ideal, then (I : J) is divisorial. Let R be a local Krull domain (e.g., a Noetherian integrally closed local domain). Then R is a discrete valuation ring if and only if the maximal ideal of R is divisorial.[2] An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.[3]
82.4 See also • divisorial sheaf
82.5 Notes [1] Bourbaki 1998, §VII.1 [2] Bourbaki Ch. VII, § 1, n. 7. Proposition 11. [3] http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdffirstpage_1&handle=euclid.rmjm/1187453107
82.6 References • Chapter 9 of Atiyah, Michael Francis; Macdonald, I.G. (1994), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 • Chapter VII.1 of Bourbaki, Nicolas (1998), Commutative algebra (2nd ed.), Springer Verlag, ISBN 3-54064239-0 • Chapter 11 of Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics 8 (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461
Chapter 83
Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its p-th power. In certain contexts it is an automorphism, but this is not true in general.
83.1 Definition Let R be a commutative ring with prime characteristic p (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism F is defined by
F (r) = rp for all r in R. Clearly this respects the multiplication of R:
F (rs) = (rs)p = rp sp = F (r)F (s) , and F(1) is clearly 1 also. What is interesting, however, is that it also respects the addition of R. The expression (r + s)p can be expanded using the binomial theorem. Because p is prime, it divides p! but not any q! for q < p; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients p! , k!(p − k)! if 1 ≤ k ≤ p − 1. Therefore the coefficients of all the terms except rp and sp are divisible by p, the characteristic, and hence they vanish.[1] Thus
F (r + s) = (r + s)p = rp + sp = F (r) + F (s) . This shows that F is a ring homomorphism. If φ : R → S is a homomorphism of rings of characteristic p, then:
ϕ(xp ) = ϕ(x)p . If FR and FS are the Frobenius endomorphisms of R and S, then this can be rewritten as:
ϕ ◦ FR = FS ◦ ϕ. 278
83.2. FIXED POINTS OF THE FROBENIUS ENDOMORPHISM
279
This means that the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic p rings to itself. If the ring R is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: F(r) = 0 means rp = 0, which by definition means that r is nilpotent of order at most p. In fact, this is an if and only if, because if r is any nilpotent, then one of its powers will be nilpotent of order at most p. In particular, if R is a field then the Frobenius endomorphism is injective. The Frobenius morphism is not necessarily surjective, even when R is a field. For example let K = Fp(t) be the finite field of p elements together with a single transcendental element; equivalently, K is the field of rational functions with coefficients in Fp. Then the image of F does not contain t. If it did, then there would be a rational function q(t)/r(t) whose p-th power q(t)p /r(t)p would equal t. But the degree of this p-th power is p deg(q) − p deg(r), which is a multiple of p. In particular, it can't be 1, which is the degree of t. This is a contradiction, so t is not in the image of F. A field K is called perfect if either it is of characteristic zero or if it is of positive characteristic and its Frobenius endomorphism is an automorphism. For example, all finite fields are perfect.
83.2 Fixed points of the Frobenius endomorphism Consider the finite field Fp. By Fermat’s little theorem, every element x of Fp satisfies xp = x. Equivalently, it is a root of the polynomial Xp − X. The elements of Fp therefore determine p roots of this equation, and because this equation has degree p it has no more than p roots over any extension. In particular, if K is an algebraic extension of Fp (such as the algebraic closure or another finite field), then Fp is the fixed field of the Frobenius automorphism of K. Let R be a ring of characteristic p > 0. If R is an integral domain, then by the same reasoning, the fixed points of Frobenius are the elements of the prime field. However, if R is not a domain, then Xp − X may have more than p roots; for example, this happens if R = Fp × Fp. A similar property is enjoyed on the finite field Fpe by the eth iterate of the Frobenius automorphism: Every element e of Fpe is a root of X p − X , so if K is an algebraic extension of Fpe and F is the Frobenius automorphism of K, then the fixed field of Fe is Fpe . If R is a domain which is an Fpe -algebra, then the fixed points of the eth iterate of Frobenius are the elements of the image of Fpe . Iterating the Frobenius map gives a sequence of elements in R:
2
3
x, xp , xp , xp , . . . . This sequence of iterates is used in defining the Frobenius closure and the tight closure of an ideal.
83.3 As a generator of Galois groups The Galois group of an extension of finite fields is generated by an iterate of the Frobenius automorphism. First, consider the case where the ground field is the prime field. Let Fq be the finite field of q elements, where q = pe . The Frobenius automorphism F of Fq fixes the prime field Fp, so it is an element of the Galois group Gal(Fq/Fp). In fact, this Galois group is cyclic and F is a generator. The order of F is e because F e acts on an element x by sending it to xq , and this is the identity on elements of Fq. Every automorphism of Fq is a power of F, and the generators are the powers F i with i coprime to e. Now consider the finite field Fq f as an extension of Fq. The Frobenius automorphism F of Fq f does not fix the ground field Fq, but its e-th iterate F e does. The Galois group Gal(Fq f /Fq) is cyclic of order f and is generated by F e . It is the subgroup of Gal(Fq f /Fp) generated by F e . The generators of Gal(Fq f /Fq) are the powers F ei where i is coprime to f. The Frobenius automorphism is not a generator of the absolute Galois group ( ) Gal Fq /Fq ,
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because this Galois group is b = lim Z/nZ, Z ←−n which is not cyclic. However, because the Frobenius automorphism is a generator of the Galois group of every finite extension of Fq, it is a generator of every finite quotient of the absolute Galois group. Consequently it is a topological generator in the usual Krull topology on the absolute Galois group.
83.4 Frobenius for schemes There are several different ways to define the Frobenius morphism for a scheme. The most fundamental is the absolute Frobenius morphism. However, the absolute Frobenius morphism behaves poorly in the relative situation because it pays no attention to the base scheme. There are several different ways of adapting the Frobenius morphism to the relative situation, each of which is useful in certain situations.
83.4.1
The absolute Frobenius morphism
Suppose that X is a scheme of characteristic p > 0. Choose an open affine subset U = Spec A of X. The ring A is an Fp-algebra, so it admits a Frobenius endomorphism. If V is an open affine subset of U, then by the naturality of Frobenius, the Frobenius morphism on U, when restricted to V, is the Frobenius morphism on V. Consequently the Frobenius morphism glues to give an endomorphism of X. This endomorphism is called the absolute Frobenius morphism of X. By definition, it is a homeomorphism of X with itself. The absolute Frobenius morphism is a natural transformation from the identity functor on the category of Fp-schemes to itself. If X is an S-scheme and the Frobenius morphism of S is the identity, then the absolute Frobenius morphism is a morphism of S-schemes. In general, however, it is not. For example, consider the ring A = Fp2 . Let X and S both equal Spec A with the structure map X → S being the identity. The Frobenius morphism on A sends a to ap . It is not a morphism of Fp2 -algebras. If it were, then multiplying by an element b in Fp2 would commute with applying the Frobenius endomorphism. But this is not true because:
b · a = ba ̸= F (b) · a = bp a. The former is the action of b in the Fp2 -algebra structure that A begins with, and the latter is the action of Fp2 induced by Frobenius. Consequently, the Frobenius morphism on Spec A is not a morphism of Fp2 -schemes. The absolute Frobenius morphism is a purely inseparable morphism of degree p. Its differential is zero. It preserves products, meaning that for any two schemes X and Y, FX×Y = FX × FY.
83.4.2
Restriction and extension of scalars by Frobenius
Suppose that φ : X → S is the structure morphism for an S-scheme X. The base scheme S has a Frobenius morphism FS. Composing φ with FS results in an S-scheme XF called the restriction of scalars by Frobenius. The restriction of scalars is actually a functor, because an S-morphism X → Y induces an S-morphism XF → YF. For example, consider a ring A of characteristic p > 0 and a finitely presented algebra over A:
R = A[X1 , . . . , Xn ]/(f1 , . . . , fm ). The action of A on R is given by:
c·
∑
aα X α =
∑
caα X α ,
where α is a multi-index. Let X = Spec R. Then XF is the affine scheme Spec R, but its structure morphism Spec R → Spec A, and hence the action of A on R, is different:
83.4. FROBENIUS FOR SCHEMES
c·
∑
aα X α =
∑
F (c)aα X α =
281
∑
cp aα X α .
Because restriction of scalars by Frobenius is simply composition, many properties of X are inherited by XF under appropriate hypotheses on the Frobenius morphism. For example, if X and SF are both finite type, then so is XF. The extension of scalars by Frobenius is defined to be:
X (p) = X ×S SF . The projection onto the S factor makes X(p) an S-scheme. If S is not clear from the context, then X(p) is denoted by X(p/S) . Like restriction of scalars, extension of scalars is a functor: An S-morphism X → Y determines an S-morphism X(p) → Y (p) . As before, consider a ring A and a finitely presented algebra R over A, and again let X = Spec R. Then:
X (p) = Spec R ⊗A AF . A global section of X(p) is of the form: ( ∑ ∑
) ⊗ bi =
aiα X α
α
i
∑∑
X α ⊗ apiα bi ,
α
i
where α is a multi-index and every aiα and bi is an element of A. The action of an element c of A on this section is:
c·
( ∑ ∑
) aiα X
α
α
i
⊗ bi =
( ∑ ∑ i
) aiα X
α
⊗ bi c.
α
Consequently, X(p) is isomorphic to: ) ( (p) (p) , Spec A[X1 , . . . , Xn ]/ f1 , . . . , fm where, if:
fj =
∑
fjβ X β ,
β
then:
(p)
fj
=
∑
p fjβ Xβ.
β
A similar description holds for arbitrary A-algebras R. Because extension of scalars is base change, it preserves limits and coproducts. This implies in particular that if X has an algebraic structure defined in terms of finite limits (such as being a group scheme), then so does X(p) . Furthermore, being a base change means that extension of scalars preserves properties such as being of finite type, finite presentation, separated, affine, and so on. Extension of scalars is well-behaved with respect to base change: Given a morphism S′ → S, there is a natural isomorphism: ′ X (p/S) ×S S ′ ∼ = (X ×S S ′ )(p/S ) .
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CHAPTER 83. FROBENIUS ENDOMORPHISM
83.4.3
Relative Frobenius
The relative Frobenius morphism of an S-scheme X is the morphism:
FX/S : X → X (p) defined by:
FX/S = (FX , 1S ). Because the absolute Frobenius morphism is natural, the relative Frobenius morphism is a morphism of S-schemes. Consider, for example, the A-algebra:
R = A[X1 , . . . , Xn ]/(f1 , . . . , fm ). We have:
(p)
(p) R(p) = A[X1 , . . . , Xn ]/(f1 , . . . , fm ).
The relative Frobenius morphism is the homomorphism R(p) → R defined by: ∑∑ i
X α ⊗ aiα 7→
α
∑∑ i
aiα X pα .
α
Relative Frobenius is compatible with base change in the sense that, under the natural isomorphism of X(p/S) ×S S′ and (X ×S S′)(p/S′) , we have:
FX/S × 1S ′ = FX×S S ′ /S ′ . Relative Frobenius is a universal homeomorphism. If X → S is an open immersion, then it is the identity. If X → S is a closed immersion determined by an ideal sheaf I of OS, then X(p) is determined by the ideal sheaf Ip and relative Frobenius is the augmentation map OS/Ip → OS/I. X is unramified over S if and only if FX/S is unramified and if and only if FX/S is a monomorphism. X is étale over S if and only if FX/S is étale and if and only if FX/S is an isomorphism.
83.4.4
Arithmetic Frobenius
See also: Arithmetic and geometric Frobenius The arithmetic Frobenius morphism of an S-scheme X is a morphism: a FX/S : X (p) → X ×S S ∼ =X
defined by:
a FX/S = 1 X × FS .
That is, it is the base change of FS by 1X.
83.4. FROBENIUS FOR SCHEMES
283
Again, if:
R = A[X1 , . . . , Xn ]/(f1 , . . . , fm ), R(p) = A[X1 , . . . , Xn ]/(f1 , . . . , fm ) ⊗A AF , then the arithmetic Frobenius is the homomorphism: ( ∑ ∑
) aiα X
α
⊗ bi 7→
α
i
∑∑ i
aiα bpi X α .
α
If we rewrite R(p) as: ( ) (p) (p) , R(p) = A[X1 , . . . , Xn ]/ f1 , . . . , fm then this homomorphism is: ∑
aα X α 7→
83.4.5
∑
apα X α .
Geometric Frobenius
Assume that the absolute Frobenius morphism of S is invertible with inverse FS−1 . Let SF −1 denote the S-scheme FS−1 : S → S . Then there is an extension of scalars of X by FS−1 :
X (1/p) = X ×S SF −1 . If:
R = A[X1 , . . . , Xn ]/(f1 , . . . , fm ), then extending scalars by FS−1 gives:
R(1/p) = A[X1 , . . . , Xn ]/(f1 , . . . , fm ) ⊗A AF −1 . If:
fj =
∑
fjβ X β ,
β
then we write:
(1/p)
fj
=
∑
1/p
fjβ X β ,
β
and then there is an isomorphism: (1/p) (1/p) R(1/p) ∼ , . . . , fm ). = A[X1 , . . . , Xn ]/(f1
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CHAPTER 83. FROBENIUS ENDOMORPHISM
The geometric Frobenius morphism of an S-scheme X is a morphism: g FX/S : X (1/p) → X ×S S ∼ =X
defined by: g FX/S = 1X × FS−1 .
It is the base change of FS−1 by 1X. Continuing our example of A and R above, geometric Frobenius is defined to be: ( ∑ ∑ i
) aiα X
α
⊗ bi 7→
α
∑∑ i
α (1/p)
After rewriting R(1/p) in terms of {fj ∑
aα X α 7→
83.4.6
∑
1/p
aiα bi X α . } , geometric Frobenius is:
α a1/p α X .
Arithmetic and geometric Frobenius as Galois actions
Suppose that the Frobenius morphism of S is an isomorphism. Then it generates a subgroup of the automorphism of group of S. If S = Spec k is the spectrum of a finite field, then its automorphism group is the Galois group of the field over the prime field, and the Frobenius morphism and its inverse are both generators of the automorphism group. In addition, X(p) and X(1/p) may be identified with X. The arithmetic and geometric Frobenius morphisms are then endomorphisms of X, and so they lead to an action of the Galois group of k on X. Consider the set of K-points X(K). This set comes with a Galois action: Each such point x corresponds to a homomorphism OX → k(x) ≅ K from the structure sheaf to the residue field at x, and the action of Frobenius on x is the application of the Frobenius morphism to the residue field. This Galois action agrees with the action of arithmetic Frobenius: The composite morphism F
OX → k(x)− →k(x) is the same as the composite morphism: a FX/S
OX −−−→OX → k(x) by the definition of the arithmetic Frobenius. Consequently, arithmetic Frobenius explicitly exhibits the action of the Galois group on points as an endomorphism of X.
83.5 Frobenius for local fields Given an unramified finite extension L/K of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.[2] Suppose L/K is an unramified extension of local fields, with ring of integers OK of K such that the residue field, the integers of K modulo their unique maximal ideal φ, is a finite field of order q. If Φ is a prime of L lying over φ, that L/K is unramified means by definition that the integers of L modulo Φ, the residue field of L, will be a finite field of order qf extending the residue field of K where f is the degree of L/K. We may define the Frobenius map for elements of the ring of integers OL of L as an automorphism sΦ of L such that sΦ (x) ≡ xq
mod Φ.
83.6. FROBENIUS FOR GLOBAL FIELDS
285
83.6 Frobenius for global fields In algebraic number theory, Frobenius elements are defined for extensions L/K of global fields that are finite Galois extensions for prime ideals Φ of L that are unramified in L/K. Since the extension is unramified the decomposition group of Φ is the Galois group of the extension of residue fields. The Frobenius element then can be defined for elements of the ring of integers of L as in the local case, by
sΦ (x) ≡ xq
mod Φ,
where q is the order of the residue field OK mod Φ. Lifts of the Frobenius are in correspondence with p-derivations.
83.7 Examples The polynomial x5 − x − 1 has discriminant 19 × 151, and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root ρ of it to the field of 3-adic numbers Q3 gives an unramified extension Q3 (ρ) of Q3 . We may find the image of ρ under the Frobenius map by locating the root nearest to ρ3 , which we may do by Newton’s method. We obtain an element of the ring of integers Z3 [ρ] in this way; this is a polynomial of degree four in ρ with coefficients in the 3-adic integers Z3 . Modulo 38 this polynomial is
ρ3 + 3(460 + 183ρ − 354ρ2 − 979ρ3 − 575ρ4 ) This is algebraic over Q and is the correct global Frobenius image in terms of the embedding of Q into Q3 ; moreover, the coefficients are algebraic and the result can be expressed algebraically. However, they are of degree 120, the order of the Galois group, illustrating the fact that explicit computations are much more easily accomplished if p-adic results will suffice. If L/K is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime φ in the base field K. For an example, consider the extension Q(β) of Q obtained by adjoining a root β satisfying
β 5 + β 4 − 4β 3 − 3β 2 + 3β + 1 = 0 to Q. This extension is cyclic of order five, with roots
2 cos 2πn 11 for integer n. It has roots which are Chebyshev polynomials of β: β2 − 2, β3 − 3β, β5 − 5β3 + 5β give the result of the Frobenius map for the primes 2, 3 and 5, and so on for larger primes not equal to 11 or of the form 22n + 1 (which split). It is immediately apparent how the Frobenius map gives a result equal mod p to the p-th power of the root β.
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CHAPTER 83. FROBENIUS ENDOMORPHISM
83.8 See also • Perfect field • Frobenioid
83.9 References [1] This is known as the Freshman’s dream. [2] Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics 27. Cambridge University Press. p. 144. ISBN 0-521-36664-X. Zbl 0744.11001.
• Hazewinkel, Michiel, ed. (2001), “Frobenius automorphism”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Hazewinkel, Michiel, ed. (2001), “Frobenius endomorphism”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 84
Function field sieve In mathematics, the function field sieve was introduced in 1994 by Leonard Adleman as an efficient technique for extracting discrete logarithms over finite fields of small characteristic, and elaborated by Adleman and Huang in 1999. Sieving for points at which a polynomial-valued function is divisible by a given polynomial is not much more difficult than sieving over the integers – the underlying structure is fairly similar, and Gray code provides a convenient way to step through multiples of a given polynomial very efficiently.
84.1 References The Adleman–Huang paper is available at Science Direct, but views the problem using very algebraic-geometric language.
287
Chapter 85
Fundamental discriminant In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax2 + bxy + cy2 is a quadratic form with integer coefficients, then D = b2 − 4ac is the discriminant of Q(x, y). Conversely, every integer D with D ≡ 0, 1 (mod 4) is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as discriminants in this theory. Every discriminant may be written as D = D0 f 2 with D0 a discriminant and f a positive integer. A discriminant D is called a fundamental discriminant if f = 1 in every such decomposition. Conversely, every discriminant D ≠ 0 can be written uniquely as D0 f 2 where D0 is a fundamental discriminant. Thus, fundamental discriminants play a similar role for discriminants as prime numbers do for all integers. There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, D is a fundamental discriminant if, and only if, one of the following statements holds • D ≡ 1 (mod 4) and is square-free, • D = 4m, where m ≡ 2 or 3 (mod 4) and m is square-free. The first ten positive fundamental discriminants are: 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 (sequence A003658 in OEIS). The first ten negative fundamental discriminants are: −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 (sequence A003657 in OEIS).
85.1 Connection with quadratic fields There is a connection between the theory of integral binary quadratic forms and the arithmetic of quadratic number fields. A basic property of this connection is that D0 is a fundamental discriminant if, and only if, D0 = 1 or D0 is the discriminant of a quadratic number field. There is exactly one quadratic field for every fundamental discriminant D0 ≠ 1, up to isomorphism. Caution: This is the reason why some authors consider 1 not to be a fundamental discriminant. One may interpret D0 = 1 as the degenerated “quadratic” field Q (the rational numbers). 288
85.2. FACTORIZATION
289
85.2 Factorization Fundamental discriminants may also be characterized by their factorization into positive and negative prime powers. Define the set
S = {−8, −4, 8, −3, 5, −7, −11, 13, 17, −19, . . .} where the prime numbers ≡ 1 (mod 4) are positive and those ≡ 3 (mod 4) are negative. Then, a number D0 ≠ 1 is a fundamental discriminant if, and only if, it is the product of pairwise relatively prime members of S.
85.3 References • Henri Cohen (1993). A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138. Berlin, New York: Springer-Verlag. ISBN 3-540-55640-0. MR 1228206. • Duncan Buell (1989). Binary quadratic forms: classical theory and modern computations. Springer-Verlag. p. 69. ISBN 0-387-97037-1. • Don Zagier (1981). Zetafunktionen und quadratische Körper. Berlin, New York: Springer-Verlag. ISBN 978-3-540-10603-6.
85.4 See also • Quadratic integer
Chapter 86
Fundamental theorem of algebra Not to be confused with Fundamental theorem of arithmetic. The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division. In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients.
86.1 History Peter Rothe, in his book Arithmetica Philosophica (published in 1608), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds “unless the equation is incomplete”, by which he meant that no coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true; for instance, he shows that the equation x4 = 4x − 3, although incomplete, has four solutions (counting multiplicities): 1 (twice), −1 + i√2, and −1 − i√2. As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degree is either 1 or 2. However, in 1702 Leibniz said that no polynomial of the type x4 + a4 (with a real and distinct from 0) can be written in such a way. Later, Nikolaus Bernoulli made the same assertion concerning the polynomial x4 − 4x3 + 2x2 + 4x + 4, but he got a letter from Euler in 1742[1] in which he was told that his polynomial happened to be equal to
(x2 − (2 + α)x + 1 +
√ √ 7 + α)(x2 − (2 − α)x + 1 + 7 − α),
where α is the square root of 4 + 2√7. Also, Euler mentioned that √ √ x4 + a4 = (x2 + a 2 · x + a2 )(x2 − a 2 · x + a2 ). A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete. Among other problems, it assumed implicitly a theorem (now known as Puiseux’s theorem) which would not be proved until more 290
86.2. PROOFS
291
than a century later, and furthermore the proof assumed the fundamental theorem of algebra. Other attempts were made by Euler (1749), de Foncenex (1759), Lagrange (1772), and Laplace (1795). These last four attempts assumed implicitly Girard’s assertion; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p(z). At the end of the 18th century, two new proofs were published which did not assume the existence of roots. One of them, due to James Wood and mainly algebraic, was published in 1798 and it was totally ignored. Wood’s proof had an algebraic gap.[2] The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap, filled by Alexander Ostrowski in 1920, as discussed in Smale 1981 (Smale writes, "...I wish to point out what an immense gap Gauss’ proof contained. It is a subtle point even today that a real algebraic plane curve cannot enter a disk without leaving. In fact even though Gauss redid this proof 50 years later, the gap remained. It was not until 1920 that Gauss’ proof was completed. In the reference Gauss, A. Ostrowski has a paper which does this and gives an excellent discussion of the problem as well...”). A rigorous proof was published by Argand in 1806; it was here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients, rather than just real coefficients. Gauss produced two other proofs in 1816 and another version of his original proof in 1849. The first textbook containing a proof of the theorem was Cauchy's Cours d'analyse de l'École Royale Polytechnique (1821). It contained Argand’s proof, although Argand is not credited for it. None of the proofs mentioned so far is constructive. It was Weierstrass who raised for the first time, in the middle of the 19th century, the problem of finding a constructive proof of the fundamental theorem of algebra. He presented his solution, that amounts in modern terms to a combination of the Durand–Kerner method with the homotopy continuation principle, in 1891. Another proof of this kind was obtained by Hellmuth Kneser in 1940 and simplified by his son Martin Kneser in 1981. Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers (which are not constructively equivalent to the Cauchy real numbers without countable choice[3] ). However, Fred Richman proved a reformulated version of the theorem that does work.[4]
86.2 Proofs All proofs below involve some analysis, or at least the topological concept of continuity of real or complex functions. Some also use differentiable or even analytic functions. This fact has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra. Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complex root. This is enough to establish the theorem in the general case because, given a non-constant polynomial p(z) with complex coefficients, the polynomial
q(z) = p(z)p(z) has only real coefficients and, if z is a zero of q(z), then either z or its conjugate is a root of p(z). A large number of non-algebraic proofs of the theorem use the fact (sometimes called “growth lemma”) that an n-th degree polynomial function p(z) whose dominant coefficient is 1 behaves like zn when |z| is large enough. A more precise statement is: there is some positive real number R such that:
1 n 2 |z |
< |p(z)| < 32 |z n |
when |z| > R.
86.2.1
Complex-analytic proofs
Find a closed disk D of radius r centered at the origin such that |p(z)| > |p(0)| whenever |z| ≥ r. The minimum of |p(z)| on D, which must exist since D is compact, is therefore achieved at some point z0 in the interior of D, but not at
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CHAPTER 86. FUNDAMENTAL THEOREM OF ALGEBRA
any point of its boundary. The Maximum modulus principle (applied to 1/p(z)) implies then that p(z0 ) = 0. In other words, z0 is a zero of p(z). A variation of this proof does not require the use of the maximum modulus principle (in fact, the same argument with minor changes also gives a proof of the maximum modulus principle for holomorphic functions). If we assume by contradiction that a := p(z0 ) ≠ 0, then, expanding p(z) in powers of z − z0 we can write
p(z) = a + ck (z − z0 )k + ck+1 (z − z0 )k+1 + . . . + cn (z − z0 )n . Here, the cj are simply the coefficients of the polynomial z → p(z + z0 ), and we let k be the index of the first coefficient following the constant term that is non-zero. But now we see that for z sufficiently close to z0 this has behavior k asymptotically similar to the simpler polynomial q(z) = a + ck (z − z0 ) , in the sense that (as is easy to check) the p(z)−q(z) function (z−z0 )k+1 is bounded by some positive constant M in some neighborhood of z0 . Therefore if we define θ0 = (arg(a) + π − arg(ck ))/k and let z = z0 + reiθ0 , then for any sufficiently small positive number r (so that the bound M mentioned above holds), using the triangle inequality we see that
|p(z)| < |q(z)| + r
k+1
p(z) − q(z) rk+1
≤ a + (−1)ck rk ei(arg(a)−arg(ck )) + M rk+1 = |a| − |ck |rk + M rk+1 . When r is sufficiently close to 0 this upper bound for |p(z)| is strictly smaller than |a|, in contradiction to the definition of z0 . (Geometrically, we have found an explicit direction θ0 such that if one approaches z0 from that direction one can obtain values p(z) smaller in absolute value than |p(z0 )|.) Another analytic proof can be obtained along this line of thought observing that, since |p(z)| > |p(0)| outside D, the minimum of |p(z)| on the whole complex plane is achieved at z0 . If |p(z0 )| > 0, then 1/p is a bounded holomorphic function in the entire complex plane since, for each complex number z, |1/p(z)| ≤ |1/p(z0 )|. Applying Liouville’s theorem, which states that a bounded entire function must be constant, this would imply that 1/p is constant and therefore that p is constant. This gives a contradiction, and hence p(z0 ) = 0. Yet another analytic proof uses the argument principle. Let R be a positive real number large enough so that every root of p(z) has absolute value smaller than R; such a number must exist because every non-constant polynomial function of degree n has at most n zeros. For each r > R, consider the number
1 2πi
∫ c(r)
p′ (z) dz, p(z)
where c(r) is the circle centered at 0 with radius r oriented counterclockwise; then the argument principle says that this number is the number N of zeros of p(z) in the open ball centered at 0 with radius r, which, since r > R, is the total number of zeros of p(z). On the other hand, the integral of n/z along c(r) divided by 2πi is equal to n. But the difference between the two numbers is
1 2πi
(
∫ c(r)
p′ (z) n − p(z) z
) dz =
1 2πi
∫ c(r)
zp′ (z) − np(z) dz. zp(z)
The numerator of the rational expression being integrated has degree at most n - 1 and the degree of the denominator is n + 1. Therefore, the number above tends to 0 as r → +∞. But the number is also equal to N − n and so N = n. Still another complex-analytic proof can be given by combining linear algebra with the Cauchy theorem. To establish that every complex polynomial of degree n > 0 has a zero, it suffices to show that every complex square matrix of size n > 0 has a (complex) eigenvalue.[5] The proof of the latter statement is by contradiction. Let A be a complex square matrix of size n > 0 and let In be the unit matrix of the same size. Assume A has no eigenvalues. Consider the resolvent function
86.2. PROOFS
293
R(z) = (zIn − A)−1 , which is a meromorphic function on the complex plane with values in the vector space of matrices. The eigenvalues of A are precisely the poles of R(z). Since, by assumption, A has no eigenvalues, the function R(z) is an entire function and Cauchy theorem implies that ∫ R(z)dz = 0. c(r)
On the other hand, R(z) expanded as a geometric series gives:
R(z) = z −1 (In − z −1 A)−1 = z −1
∞ ∑ 1 k A · zk
k=0
This formula is valid outside the closed disc of radius ||A|| (the operator norm of A). Let r > ||A||. Then ∫ R(z)dz = c(r)
∞ ∫ ∑ k=0
c(r)
dz z k+1
Ak = 2πiIn
(in which only the summand k = 0 has a nonzero integral). This is a contradiction, and so A has an eigenvalue. Finally, Rouché's theorem gives perhaps the shortest proof of the theorem.
86.2.2
Topological proofs
Let z0 ∈ C be such that the minimum of |p(z)| on the whole complex plane is achieved at z0 ; it was seen at the proof which uses Liouville’s theorem that such a number must exist. We can write p(z) as a polynomial in z − z0 : there is some natural number k and there are some complex numbers ck, ck ₊ ₁, ..., cn such that ck ≠ 0 and that
p(z) = p(z0 ) + ck (z − z0 )k + ck+1 (z − z0 )k+1 + · · · + cn (z − z0 )n . In the case that p(z0 ) is nonzero, it follows that if a is a kth root of −p(z0 )/ck and if t is positive and sufficiently small, then |p(z0 + ta)| < |p(z0 )|, which is impossible, since |p(z0 )| is the minimum of |p| on D. For another topological proof by contradiction, suppose that p(z) has no zeros. Choose a large positive number R such that, for |z| = R, the leading term zn of p(z) dominates all other terms combined; in other words, such that |z|n > |an₋₁zn−1 + ··· + a0 |. As z traverses the circle given by the equation |z| = R once counter-clockwise, p(z), like zn , winds n times counter-clockwise around 0. At the other extreme, with |z| = 0, the “curve” p(z) is simply the single (nonzero) point p(0), whose winding number is clearly 0. If the loop followed by z is continuously deformed between these extremes, the path of p(z) also deforms continuously. We can explicitly write such a deformation as H(Reiθ ,t) = p((1 − t)Reiθ ), where 0 ≤ t ≤ 1. If one views the variable t as time, then at time zero the curve is p(z) and at time one the curve is p(0). Clearly at every point t, p(z) cannot be zero by the original assumption, therefore during the deformation, the curve never crosses zero. Therefore the winding number of the curve around zero should never change. However, given that the winding number started as n and ended as 0, this is absurd. Therefore, p(z) has at least one zero.
86.2.3
Algebraic proofs
These proofs use two facts about real numbers that require only a small amount of analysis (more precisely, the intermediate value theorem): • every polynomial with odd degree and real coefficients has some real root;
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CHAPTER 86. FUNDAMENTAL THEOREM OF ALGEBRA
• every non-negative real number has a square root. The second fact, together with the quadratic formula, implies the theorem for real quadratic polynomials. In other words, algebraic proofs of the fundamental theorem actually show that if R is any real-closed field, then its extension C = R(√−1) is algebraically closed. As mentioned above, it suffices to check the statement “every non-constant polynomial p(z) with real coefficients has a complex root”. This statement can be proved by induction on the greatest non-negative integer k such that 2k divides the degree n of p(z). Let a be the coefficient of zn in p(z) and let F be a splitting field of p(z) over C; in other words, the field F contains C and there are elements z1 , z2 , ..., zn in F such that p(z) = a(z − z1 )(z − z2 ) · · · (z − zn ). If k = 0, then n is odd, and therefore p(z) has a real root. Now, suppose that n = 2k m (with m odd and k > 0) and that the theorem is already proved when the degree of the polynomial has the form 2k − 1 m′ with m′ odd. For a real number t, define:
qt (z) =
∏
(z − zi − zj − tzi zj ) .
1≤i
Then the coefficients of qt(z) are symmetric polynomials in the zi's with real coefficients. Therefore, they can be expressed as polynomials with real coefficients in the elementary symmetric polynomials, that is, in −a1 , a2 , ..., (−1)n an. So qt(z) has in fact real coefficients. Furthermore, the degree of qt(z) is n(n − 1)/2 = 2k−1 m(n − 1), and m(n − 1) is an odd number. So, using the induction hypothesis, qt has at least one complex root; in other words, zi + zj + tzizj is complex for two distinct elements i and j from {1, ..., n}. Since there are more real numbers than pairs (i, j), one can find distinct real numbers t and s such that zi + zj + tzizj and zi + zj + szizj are complex (for the same i and j). So, both zi + zj and zizj are complex numbers. It is easy to check that every complex number has a complex square root, thus every complex polynomial of degree 2 has a complex root by the quadratic formula. It follows that zi and zj are complex numbers, since they are roots of the quadratic polynomial z2 − (zi + zj)z + zizj. J. Shipman showed in 2007 that the assumption that odd degree polynomials have roots is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed (so “odd” can be replaced by “odd prime” and furthermore this holds for fields of all characteristics). For axiomatization of algebraically closed fields, this is the best possible, as there are counterexamples if a single prime is excluded. However, these counterexamples rely on −1 having a square root. If we take a field where −1 has no square root, and every polynomial of degree n ∈ I has a root, where I is any fixed infinite set of odd numbers, then every polynomial f(x) of odd degree has a root (since (x2 + 1)k f(x) has a root, where k is chosen so that deg(f) + 2k ∈ I). Another algebraic proof of the fundamental theorem can be given using Galois theory. It suffices to show that C has no proper finite field extension.[6] Let K/C be a finite extension. Since the normal closure of K over R still has a finite degree over C (or R), we may assume without loss of generality that K is a normal extension of R (hence it is a Galois extension, as every algebraic extension of a field of characteristic 0 is separable). Let G be the Galois group of this extension, and let H be a Sylow 2-subgroup of G, so that the order of H is a power of 2, and the index of H in G is odd. By the fundamental theorem of Galois theory, there exists a subextension L of K/R such that Gal(K/L) = H. As [L:R] = [G:H] is odd, and there are no nonlinear irreducible real polynomials of odd degree, we must have L = R, thus [K:R] and [K:C] are powers of 2. Assuming by way of contradiction that [K:C] > 1, we conclude that the 2-group Gal(K/C) contains a subgroup of index 2, so there exists a subextension M of C of degree 2. However, C has no extension of degree 2, because every quadratic complex polynomial has a complex root, as mentioned above. This shows that [K:C] = 1, and therefore K = C, which completes the proof.
86.2.4
Geometric proofs
There exists still another way to approach the fundamental theorem of algebra, due to J. M. Almira and A. Romero: by Riemannian Geometric arguments. The main idea here is to prove that the existence of a non-constant polynomial p(z) without zeros implies the existence of a flat Riemannian metric over the sphere S2 . This leads to a contradiction, since the sphere is not flat. Recall that a Riemannian surface (M, g) is said to be flat if its Gaussian curvature, which we denote by Kg, is identically null. Now, Gauss–Bonnet theorem, when applied to the sphere S2 , claims that
86.3. COROLLARIES
295
∫ Kg = 4π S2
which proves that the sphere is not flat. Let us now assume that n > 0 and p(z) = a0 + a1 z + ⋅⋅⋅ + anzn ≠ 0 for each complex number z. Let us define p*(z) = zn p(1/z) = a0 zn + a1 zn−1 + ⋅⋅⋅ + an. Obviously, p*(z) ≠ 0 for all z in C. Consider the polynomial f(z) = p(z)p*(z). Then f(z) ≠ 0 for each z in C. Furthermore,
f ( w1 ) = p( w1 )p∗ ( w1 ) = w−2n p∗ (w)p(w) = w−2n f (w) We can use this functional equation to prove that g, given by
g=
1 2
|f (w)| n
|dw|2
for w in C, and
g=
1 2
|f (1/w)| n
|d(1/w)|2
for w ∈ S2 \{0}, is a well defined Riemannian metric over the sphere S2 (which we identify with the extended complex plane C ∪ {∞}). Now, a simple computation shows that
∀w ∈ C :
1 |f (w)|
1 n
Kg =
1 1 ∆ log |f (w)| = ∆Re(log f (w)) = 0 n n
since the real part of an analytic function is harmonic. This proves that Kg = 0.
86.3 Corollaries Since the fundamental theorem of algebra can be seen as the statement that the field of complex numbers is algebraically closed, it follows that any theorem concerning algebraically closed fields applies to the field of complex numbers. Here are a few more consequences of the theorem, which are either about the field of real numbers or about the relationship between the field of real numbers and the field of complex numbers: • The field of complex numbers is the algebraic closure of the field of real numbers. • Every polynomial in one variable z with complex coefficients is the product of a complex constant and polynomials of the form z + a with a complex. • Every polynomial in one variable x with real coefficients can be uniquely written as the product of a constant, polynomials of the form x + a with a real, and polynomials of the form x2 + ax + b with a and b real and a2 − 4b < 0 (which is the same thing as saying that the polynomial x2 + ax + b has no real roots). This implies that the number of non-real complex roots (up to multiplicity) is always even. • Every rational function in one variable x, with real coefficients, can be written as the sum of a polynomial function with rational functions of the form a/(x − b)n (where n is a natural number, and a and b are real numbers), and rational functions of the form (ax + b)/(x2 + cx + d)n (where n is a natural number, and a, b, c, and d are real numbers such that c2 − 4d < 0). A corollary of this is that every rational function in one variable and real coefficients has an elementary primitive. • Every algebraic extension of the real field is isomorphic either to the real field or to the complex field.
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86.4 Bounds on the zeros of a polynomial Main article: Properties of polynomial roots While the fundamental theorem of algebra states a general existence result, it is of some interest, both from the theoretical and from the practical point of view, to have information on the location of the zeros of a given polynomial. The simpler result in this direction is a bound on the modulus: all zeros ζ of a monic polynomial zn +an−1 zn−1 +···+ a1 z+a0 satisfy an inequality |ζ| ≤ R∞, where
R∞ := 1 + max{|a0 |, · · · , |an−1 |}. Notice that, as stated, this is not yet an existence result but rather an example of what is called an a priori bound: it says that if there are solutions then they lie inside the closed disk of center the origin and radius R∞. However, once coupled with the fundamental theorem of algebra it says that the disk contains in fact at least one solution. More generally, a bound can be given directly in terms of any p-norm of the n-vector of coefficients a:=( a0 , a1 ,..., an−1 ) , that is |ζ| ≤ Rp, where Rp is precisely the q-norm of the 2-vector (1,∥a∥p ) , q being the conjugate exponent of p, 1/p + 1/q = 1, for any 1 ≤ p ≤ ∞. Thus, the modulus of any solution is also bounded by { } ∑ R1 := max 1, |ak | , 0≤k
[ Rp := 1 +
( ∑
|ak |
p
) pq ] q1
,
0≤k
for 1 < p < ∞, and in particular
R2 :=
√ ∑
|ak |2
0≤k≤n
(where we define an to mean 1, which is reasonable since 1 is indeed the n-th coefficient of our polynomial). The case of a generic polynomial of degree n, P (z):=an zn +an−1 zn−1 +···+a1 z+a0 , is of course reduced to the case of a monic, dividing all coefficients by an ≠ 0. Also, in case that 0 is not a root, i.e. a0 ≠ 0., bounds from below on the roots ζ follow immediately as bounds from above on ζ1 , that is, the roots of a0 zn +a1 zn−1 +···+an−1 z+an . Finally, the distance |ζ − ζ0 | from the roots ζ to any point ζ0 can be estimated from below and above, seeing ζ − ζ0 as zeros of the polynomial P (z + ζ0 ) , whose coefficients are the Taylor expansion of P(z) at z = ζ0 . We report here the proof of the above bounds, which is short and elementary. Let ζ be a root of the polynomial z n +an−1 z n−1 +···+a1 z+a0 ; in order to prove the inequality |ζ| ≤ Rp we can assume, of course, |ζ| > 1. Writing the equation as −ζ n =an−1 ζ n−1 +···+a1 ζ+a0 , and using the Hölder’s inequality we find |ζ|n ≤∥a∥p ∥(ζ n−1 ,··· ,ζ,1)∥q . Now, if p = 1, this is |ζ|n ≤∥a∥1 max{|ζ|n−1 ,··· ,|ζ|,1}=∥a∥1 |ζ|n−1 , thus |ζ|≤max{1,∥a∥1 } . In the case 1 < p ≤ ∞, taking into account the summation formula for a geometric progression, we have (
|ζ| ≤ ∥a∥p |ζ| n
q(n−1)
+ · · · + |ζ| + 1 q
)1/q
( = ∥a∥p
|ζ|qn − 1 |ζ|q − 1
)1/q
( ≤ ∥a∥p
|ζ|qn |ζ|q − 1
)1/q ,
qn
|ζ| q q thus |ζ|nq ≤∥a∥qp |ζ| q −1 and simplifying, |ζ| ≤1+∥a∥p . Therefore |ζ|≤∥(1,∥a∥p )∥q =Rp holds, for all 1 ≤ p ≤ ∞.
86.5 Notes • See section Le rôle d'Euler in C. Gilain’s article Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral. • Concerning Wood’s proof, see the article A forgotten paper on the fundamental theorem of algebra, by Frank Smithies.
86.6. REFERENCES
297
• For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; A weak countable choice principle; available from [1]. • See Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; available from [2]. • A proof of the fact that this suffices can be seen here.
86.6 References [1] See section Le rôle d'Euler in C. Gilain’s article Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral. [2] Concerning Wood’s proof, see the article A forgotten paper on the fundamental theorem of algebra, by Frank Smithies. [3] For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman; 1998; A weak countable choice principle; available from . [4] See Fred Richman; 1998; The fundamental theorem of algebra: a constructive development without choice; available from . [5] A proof of the fact that this suffices can be seen here. [6] A proof of the fact that this suffices can be seen here.
86.6.1
Historic sources
• Cauchy, Augustin Louis (1821), Cours d'Analyse de l'École Royale Polytechnique, 1ère partie: Analyse Algébrique, Paris: Éditions Jacques Gabay (published 1992), ISBN 2-87647-053-5 (tr. Course on Analysis of the Royal Polytechnic Academy, part 1: Algebraic Analysis) • Euler, Leonhard (1751), “Recherches sur les racines imaginaires des équations”, Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin (Berlin) 5: 222–288. English translation: Euler, Leonhard (1751), “Investigations on the Imaginary Roots of Equations” (PDF), Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin (Berlin) 5: 222–288 • Gauss, Carl Friedrich (1799), Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse, Helmstedt: C. G. Fleckeisen (tr. New proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree). • Gauss, Carl Friedrich (1866), Carl Friedrich Gauss Werke, Band III, Königlichen Gesellschaft der Wissenschaften zu Göttingen 1. Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (1799), pp.1-31., p. 1, at Google Books - first proof. 2. Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (1815 Dec), pp.32-56., p. 32, at Google Books second proof. 3. Theorematis de resolubilitate functionum algebraicarum integrarum in factores reales demonstratio tertia Supplementum commentationis praecedentis (1816 Jan), pp.57-64., p. 57, at Google Books - third proof. 4. Beiträge zur Theorie der algebraischen Gleichungen (1849 Juli), pp.71-103., p. 71, at Google Books fourth proof. • Kneser, Hellmuth (1940), “Der Fundamentalsatz der Algebra und der Intuitionismus”, Mathematische Zeitschrift 46: 287–302, doi:10.1007/BF01181442, ISSN 0025-5874 (The Fundamental Theorem of Algebra and Intuitionism). • Kneser, Martin (1981), “Ergänzung zu einer Arbeit von Hellmuth Kneser über den Fundamentalsatz der Algebra”, Mathematische Zeitschrift 177 (2): 285–287, doi:10.1007/BF01214206, ISSN 0025-5874 (tr. An extension of a work of Hellmuth Kneser on the Fundamental Theorem of Algebra).
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• Ostrowski, Alexander (1920), "Über den ersten und vierten Gaußschen Beweis des Fundamental-Satzes der Algebra”, Carl Friedrich Gauss Werke Band X Abt. 2 (tr. On the first and fourth Gaussian proofs of the Fundamental Theorem of Algebra). • Weierstraß, Karl (1891). “Neuer Beweis des Satzes, dass jede ganze rationale Function einer Veränderlichen dargestellt werden kann als ein Product aus linearen Functionen derselben Veränderlichen”. Sitzungsberichte der königlich preussischen Akademie der Wissenschaften zu Berlin. pp. 1085–1101. (tr. New proof of the theorem that every integral rational function of one variable can be represented as a product of linear functions of the same variable).
86.6.2
Recent literature
• Almira, J.M.; Romero, A. (2007), “Yet another application of the Gauss-Bonnet Theorem for the sphere”, Bulletin of the Belgian Mathematical Society 14: 341–342 • Almira, J.M.; Romero, A. (2012), “Some Riemannian geometric proofs of the Fundamental Theorem of Algebra” (PDF), Differential Geometry - Dynamical Systems 14: 1–4 • de Oliveira, O.R.B. (2011), “The Fundamental Theorem of Algebra: an elementary and direct proof”, Mathematical Intelligencer 33 (2): 1–2 • de Oliveira, O.R.B. (2012), “The Fundamental Theorem of Algebra: from the four basic operations”, American Mathematical Monthly 119 (9): 753–758 • Fine, Benjamin; Rosenberger, Gerhard (1997), The Fundamental Theorem of Algebra, Undergraduate Texts in Mathematics, Berlin: Springer-Verlag, ISBN 978-0-387-94657-3, MR 1454356 • Gersten, S.M.; Stallings, John R. (1988), “On Gauss’s First Proof of the Fundamental Theorem of Algebra”, Proceedings of the AMS 103 (1): 331–332, doi:10.2307/2047574, ISSN 0002-9939, JSTOR 2047574 • Gilain, Christian (1991), “Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral”, Archive for History of Exact Sciences 42 (2): 91–136, doi:10.1007/BF00496870, ISSN 0003-9519 (tr. On the history of the fundamental theorem of algebra: theory of equations and integral calculus.) • Netto, Eugen; Le Vavasseur, Raymond (1916), “Les fonctions rationnelles §80–88: Le théorème fondamental”, in Meyer, François; Molk, Jules, Encyclopédie des Sciences Mathématiques Pures et Appliquées, tome I, vol. 2, Éditions Jacques Gabay (published 1992), ISBN 2-87647-101-9 (tr. The rational functions §80–88: the fundamental theorem). • Remmert, Reinhold (1991), “The Fundamental Theorem of Algebra”, in Ebbinghaus, Heinz-Dieter; Hermes, Hans; Hirzebruch, Friedrich, Numbers, Graduate Texts in Mathematics 123, Berlin: Springer-Verlag, ISBN 978-0-387-97497-2 • Shipman, Joseph (2007), “Improving the Fundamental Theorem of Algebra”, Mathematical Intelligencer 29 (4): 9–14, doi:10.1007/BF02986170, ISSN 0343-6993 • Smale, Steve (1981), “The Fundamental Theorem of Algebra and Complexity Theory”, Bulletin (new series) of the American Mathematical Society 4 (1) • Smith, David Eugene (1959), A Source Book in Mathematics, Dover, ISBN 0-486-64690-4 • Smithies, Frank (2000), “A forgotten paper on the fundamental theorem of algebra”, Notes & Records of the Royal Society 54 (3): 333–341, doi:10.1098/rsnr.2000.0116, ISSN 0035-9149 • Taylor, Paul (2 June 2007), Gauss’s second proof of the fundamental theorem of algebra - English translation of Gauss’s second proof. • van der Waerden, Bartel Leendert (2003), Algebra I (7th ed.), Springer-Verlag, ISBN 0-387-40624-7
86.7. EXTERNAL LINKS
299
86.7 External links • Algebra, fundamental theorem of at Encyclopaedia of Mathematics • Fundamental Theorem of Algebra — a collection of proofs • D. J. Velleman: The Fundamental Theorem of Algebra: A Visual Approach, PDF (unpublished paper), visualisation of d'Alembert’s, Gauss’s and the winding number proofs • Fundamental Theorem of Algebra Module by John H. Mathews • Bibliography for the Fundamental Theorem of Algebra • From the Fundamental Theorem of Algebra to Astrophysics: A “Harmonious” Path • Gauss’s first proof (in Latin) at Google Books • Gauss’s first proof (in Latin) at Google Books
Chapter 87
Fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. In its most basic form, the theorem asserts that given a field extension E/F that is finite and Galois, there is a one-toone correspondence between its intermediate fields and subgroups of its Galois group. (Intermediate fields are fields K satisfying F ⊆ K ⊆ E; they are also called subextensions of E/F.)
87.1 Explicit description of the correspondence For finite extensions, the correspondence can be described explicitly as follows. • For any subgroup H of Gal(E/F), the corresponding fixed field, denoted EH , is the set of those elements of E which are fixed by every automorphism in H. • For any intermediate field K of E/F, the corresponding subgroup is Aut(E/K), that is, the set of those automorphisms in Gal(E/F) which fix every element of K. The fundamental theorem says that this correspondence is a one-to-one correspondence if (and only if) E/F is a Galois extension. For example, the topmost field E corresponds to the trivial subgroup of Gal(E/F), and the base field F corresponds to the whole group Gal(E/F). The notation Gal(E/F) is only used for Galois extensions. If E/F is Galois, then Gal(E/F) = Aut(E/F). If E/F is not Galois, then the “correspondence” gives only an injective (but not surjective) map from { subgroups of Aut(E/F) } to { subfields of E/F } , and a surjective (but not injective) map in the reverse direction. In particular, if E/F is not Galois, then F is not the fixed field of any subgroup of Aut(E/F).
87.2 Properties of the correspondence The correspondence has the following useful properties. • It is inclusion-reversing. The inclusion of subgroups H1 EH2 holds.
H2 holds if and only if the inclusion of fields EH1 ⊇
• Degrees of extensions are related to orders of groups, in a manner consistent with the inclusion-reversing property. Specifically, if H is a subgroup of Gal(E/F), then |H| = [E:EH ] and |Gal(E/F)/H| = [EH :F]. • The field EH is a normal extension of F (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H is a normal subgroup of Gal(E/F). In this case, the restriction of the elements of Gal(E/F) to EH induces an isomorphism between Gal(EH /F) and the quotient group Gal(E/F)/H. 300
87.3. EXAMPLE 1
301
87.3 Example 1
Lattice of subgroups and subfields
Consider the field K = Q(√2,√3) = [Q(√2)](√3). Since K is first determined by adjoining √2, then √3, each element of K can be written as: (a + b√2) + (c +d√2)√3, where a, b, c and d are rational numbers. Its Galois group G = Gal(K/Q) can be determined by examining the automorphisms of K which fix a. Each such automorphism must send √2 to either √2 or –√2, and must send √3 to either √3 or –√3 since the permutations in a Galois group can only permute the roots of an irreducible polynomial. Suppose that f exchanges √2 and –√2, so ( √ √ √ ) √ √ √ √ √ √ f (a + b 2) + (c + d 2) 3 = (a − b 2) + (c − d 2) 3 = a − b 2 + c 3 − d 6, and g exchanges √3 and –√3, so ( √ √ √ ) √ √ √ √ √ √ g (a + b 2) + (c + d 2) 3 = (a + b 2) − (c + d 2) 3 = a + b 2 − c 3 − d 6. These are clearly automorphisms of K. There is also the identity automorphism e which does not change anything, and the composition of f and g which changes the signs on both radicals: ( √ √ √ ) √ √ √ √ √ √ (f g) (a + b 2) + (c + d 2) 3 = (a − b 2) − (c − d 2) 3 = a − b 2 − c 3 + d 6. Therefore G = {1, f, g, f g} , and G is isomorphic to the Klein four-group. It has five subgroups, each of which correspond via the theorem to a subfield of K. • The trivial subgroup (containing only the identity element) corresponds to all of K. • The entire group G corresponds to the base field Q. • The two-element subgroup {1, f} corresponds to the subfield Q(√3), since f fixes √3. • The two-element subgroup {1, g} corresponds to the subfield Q(√2), again since g fixes √2. • The two-element subgroup {1, fg} corresponds to the subfield Q(√6), since fg fixes √6.
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CHAPTER 87. FUNDAMENTAL THEOREM OF GALOIS THEORY
87.4 Example 2
Lattice of subgroups and subfields
The following is the simplest case where the Galois group is not abelian. Consider the splitting field K of the polynomial x3 −2 over Q; that is, K = Q (θ, ω), where θ is a cube root of 2, and ω is a cube root of 1 (but not 1 itself). For example, if we imagine K to be inside the field of complex numbers, we may take θ to be the real cube root of 2, and ω to be √ −1 3 ω= +i . 2 2 It can be shown that the Galois group G = Gal(K/Q) has six elements, and is isomorphic to the group of permutations of three objects. It is generated by (for example) two automorphisms, say f and g, which are determined by their effect on θ and ω,
f (θ) = ωθ, g(θ) = θ,
f (ω) = ω, g(ω) = ω 2 ,
and then { } G = 1, f, f 2 , g, gf, gf 2 . The subgroups of G and corresponding subfields are as follows: • As usual, the entire group G corresponds to the base field Q, and the trivial group {1} corresponds to the whole field K. • There is a unique subgroup of order 3, namely {1, f, f 2 }. The corresponding subfield is Q(ω), which has degree two over Q (the minimal polynomial of ω is x2 + x + 1), corresponding to the fact that the subgroup has index two in G. Also, this subgroup is normal, corresponding to the fact that the subfield is normal over Q. • There are three subgroups of order 2, namely {1, g}, {1, gf } and {1, gf 2 }, corresponding respectively to the three subfields Q(θ), Q(ωθ), Q(ω2 θ). These subfields have degree three over Q, again corresponding to the subgroups having index 3 in G. Note that the subgroups are not normal in G, and this corresponds to the fact that the subfields are not Galois over Q. For example, Q(θ) contains only a single root of the polynomial x3 −2, so it cannot be normal over Q.
87.5. EXAMPLE 3
303
87.5 Example 3 Let E = Q(λ) be the field of rational functions in λ and let
G=
{ λ,
} 1 λ−1 1 λ , , , , 1 − λ ⊂ Aut(E) 1−λ λ λ λ−1
which is a group under composition, isomorphic to S3 (see: six cross-ratios). Let F be the fixed field of G , then Gal(E/F ) = G . If H is a subgroup of G then the coefficients of the following polynomial
P (T ) :=
∏
(T − h) ∈ E[T ]
h∈H
generate the fixed field of H . Galois correspondence means that every subfield of E/F can be constructed this way. For example, if H = {λ, 1 − λ} then the fixed field is Q(λ(1 − λ)) and if H = {λ, 1/λ} then the fixed field is Q(λ+1/λ) . Likewise, one can write F , the fixed field of G , as Q(j) with j as in J-invariant#Alternate Expressions. Similar examples can be constructed for each of the symmetry groups of the platonic solids as these also have faithful actions on the projective line P 1 (C) and hence on C(x) .
87.6 Applications The theorem classifies the intermediate fields of E/F in terms of group theory. This translation between intermediate fields and subgroups is key to showing that the general quintic equation is not solvable by radicals (see Abel–Ruffini theorem). One first determines the Galois groups of radical extensions (extensions of the form F(α) where α is an n-th root of some element of F), and then uses the fundamental theorem to show that solvable extensions correspond to solvable groups. Theories such as Kummer theory and class field theory are predicated on the fundamental theorem.
87.7 Infinite case There is also a version of the fundamental theorem that applies to infinite algebraic extensions, which are normal and separable. It involves defining a certain topological structure, the Krull topology, on the Galois group; only subgroups that are also closed sets are relevant in the correspondence.
87.8 References
Chapter 88
Fundamental unit (number theory) In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet’s unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units.[1] Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. Neukirch 1999, p. 42).
88.1 Real quadratic fields √ For the real quadratic field K = Q( d) (with d square-free), the fundamental unit ε is commonly normalized so that ε > 1 (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the discriminant of K, then the fundamental unit is √ a+b ∆ ε= 2 where (a, b) is the smallest solution to[2]
x2 − ∆y 2 = ±4 in positive integers. This equation is basically Pell’s equation√or the negative Pell equation and its solutions can be obtained similarly using the continued fraction expansion of ∆ . Whether or not x2 − Δy2 = −4 has a solution determines whether or not the class group of K is the same as its narrow class group, or equivalently, whether or not there is a unit of norm −1 √ in K. This equation is known to have a solution if, and only if, the period of the continued fraction expansion of ∆ is odd. A simpler relation can be obtained using congruences: if Δ is divisible by a prime that is congruent to 3 modulo 4, then K does not have a unit of norm −1. However, the converse does not hold as shown by the example d = 34.[3] In the early 1990s, Peter Stevenhagen proposed a probabilistic model that led him to a conjecture on how often the converse fails. Specifically, if D(X) is the number of real quadratic fields whose discriminant Δ < X is not divisible by a prime congruent to 3 modulo 4 and D− (X) is those who have a unit of norm −1, then[4] ∏ ( ) D− (x) =1− 1 − 2−j . X→∞ D(x) lim
j≥1odd
In other words, the converse fails about 42% of the time. As of March 2012, a recent result towards this conjecture was provided by Étienne Fouvry and Jürgen Klüners[5] who show that the converse fails between 33% and 59% of the time. 304
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88.2 Cubic fields If K is a complex cubic field then it has a unique real embedding and the fundamental unit ε can be picked uniquely such that |ε| > 1 in this embedding. If the discriminant Δ of K satisfies |Δ| ≥ 33, then[6]
ϵ3 >
|∆| − 27 . 4
√ √ √ 3 For example, the fundamental unit of Q( 3 2) is 1 + 3 2 + 22 whose cube is ≈ 56.9, whereas the discriminant of this field is −108 and |∆| − 27 = 20.25. 4
88.3 Notes [1] Alaca & Williams 2004, §13.4 [2] Neukirch 1999, Exercise I.7.1 [3] Alaca & Williams 2004, Table 11.5.4 [4] Stevenhagen 1993, Conjecture 1.4 [5] Fouvry & Klüners 2010 [6] Alaca & Williams 2004, Theorem 13.6.1
88.4 References • Alaca, Şaban; Williams, Kenneth S. (2004), Introductory algebraic number theory, Cambridge University Press, ISBN 978-0-521-54011-7 • Duncan Buell (1989). Binary quadratic forms: classical theory and modern computations. Springer-Verlag. pp. 92–93. ISBN 0-387-97037-1. • Fouvry, Étienne; Klüners, Jürgen (2010), “On the negative Pell equation”, Annals of Mathematics 2 (3): 2035– 2104, doi:10.4007/annals.2010.172.2035, MR 2726105 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Stevenhagen, Peter (1993), “The number of real quadratic fields having units of negative norm”, Experimental Mathematics 2 (2): 121–136, doi:10.1080/10586458.1993.10504272, MR 1259426
88.5 External links • Weisstein, Eric W., “Fundamental Unit”, MathWorld.
Chapter 89
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.
89.1 History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so that this was a non-abelian theory. It was formulated abstractly as a theory of class formations. Two developments of the 1960s turned the position around. Firstly, Galois cohomology appeared as the foundational layer of étale cohomology theory (roughly speaking, the theory as it applies to zero-dimensional schemes). Secondly, non-abelian class field theory was launched as part of the Langlands philosophy. The earliest results identifiable as Galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves. The normal basis theorem implies that the first cohomology group of the additive group of L will vanish; this is a result on general field extensions, but was known in some form to Richard Dedekind. The corresponding result for the multiplicative group is known as Hilbert’s Theorem 90, and was known before 1900. Kummer theory was another such early part of the theory, giving a description of the connecting homomorphism coming from the m-th power map. In fact for a while the multiplicative case of a 1-cocycle for groups that are not necessarily cyclic was formulated as the solubility of Noether’s equations, named for Emmy Noether; they appear under this name in Emil Artin's treatment of Galois theory, and may have been folklore in the 1920s. The case of 2-cocycles for the multiplicative group is that of the Brauer group, and the implications seem to have been well known to algebraists of the 1930s. In another direction, that of torsors, these were already implicit in the infinite descent arguments of Fermat for elliptic curves. Numerous direct calculations were done, and the proof of the Mordell–Weil theorem had to proceed by some surrogate of a finiteness proof for a particular H 1 group. The 'twisted' nature of objects over fields that are not algebraically closed, which are not isomorphic but become so over the algebraic closure, was also known in many cases linked to other algebraic groups (such as quadratic forms, simple algebras, Severi–Brauer varieties), in the 1930s, before the general theory arrived. The needs of number theory were in particular expressed by the requirement to have control of a local-global principle for Galois cohomology. This was formulated by means of results in class field theory, such as Hasse’s norm theorem. In the case of elliptic curves it led to the key definition of the Tate–Shafarevich group in the Selmer group, which is the obstruction to the success of a local-global principle. Despite its great importance, for example in the Birch and Swinnerton-Dyer conjecture, it proved very difficult to get any control of it, until results of Karl Rubin gave a way to show in some cases it was finite (a result generally believed, since its conjectural order was predicted by an L-function formula). 306
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The other major development of the theory, also involving John Tate was the Tate–Poitou duality result. Technically speaking, G may be a profinite group, in which case the definitions need to be adjusted to allow only continuous cochains.
89.2 References • Serre, Jean-Pierre (2002), Galois cohomology, Springer Monographs in Mathematics, Translated from the French by Patrick Ion, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42192-4, MR 1867431, Zbl 1004.12003, translation of Cohomologie Galoisienne, Springer-Verlag Lecture Notes 5 (1964). • Milne, James S. (2006), Arithmetic duality theorems (2nd ed.), Charleston, SC: BookSurge, LLC, ISBN 9781-4196-4274-6, MR 2261462, Zbl 1127.14001 • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, Zbl 0948.11001, MR 1737196
Chapter 90
Galois extension In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. [1] A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
90.1 Characterization of Galois extensions An important theorem of Emil Artin states that for a finite extension E/F, each of the following statements is equivalent to the statement that E/F is Galois: • E/F is a normal extension and a separable extension. • E is a splitting field of a separable polynomial with coefficients in F. • |Aut(E/F)| = [E:F], that is, the number of automorphisms equals the degree of the extension. Other equivalent statements are: • Every irreducible polynomial in F[x] with at least one root in E splits over E and is separable. • |Aut(E/F)| ≥ [E:F], that is, the number of automorphisms is at least the degree of the extension. • F is the fixed field of a subgroup of Aut(E). • F is the fixed field of Aut(E/F). • There is a one-to-one correspondence between subfields of E/F and subgroups of Aut(E/F).
90.2 Examples There are two basic ways to construct examples of Galois extensions. • Take any field E, any subgroup of Aut(E), and let F be the fixed field. • Take any field F, any separable polynomial in F[x], and let E be its splitting field. Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cube root of 2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first 308
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of them is the splitting field of x2 − 2; the second has normal closure that includes the complex cube roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and x3 − 2 has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory ¯ of an arbitrary field K is Galois over K if and only if K is a perfect field. An algebraic closure K
90.3 References [1] See the article Galois group for definitions of some of these terms and some examples.
90.4 See also • Artin, Emil (1998). Galois Theory. Edited and with a supplemental chapter by Arthur N. Milgram. Mineola, NY: Dover Publications. ISBN 0-486-62342-4. MR 1616156. • Bewersdorff, Jörg (2006). Galois theory for beginners. Student Mathematical Library 35. Translated from the second German (2004) edition by David Kramer. American Mathematical Society. ISBN 0-8218-3817-2. MR 2251389. • Edwards, Harold M. (1984). Galois Theory. Graduate Texts in Mathematics 101. New York: Springer-Verlag. ISBN 0-387-90980-X. MR 0743418. (Galois’ original paper, with extensive background and commentary.) • Funkhouser, H. Gray (1930). “A short account of the history of symmetric functions of roots of equations”. American Mathematical Monthly (The American Mathematical Monthly, Vol. 37, No. 7) 37 (7): 357–365. doi:10.2307/2299273. JSTOR 2299273. • Hazewinkel, Michiel, ed. (2001), “Galois theory”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4 • Jacobson, Nathan (1985). Basic Algebra I (2nd ed.). W.H. Freeman and Company. ISBN 0-7167-1480-9. (Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.) • Janelidze, G.; Borceux, Francis (2001). Galois theories. Cambridge University Press. ISBN 978-0-521-803090. (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.) • Lang, Serge (1994). Algebraic Number Theory. Graduate Texts in Mathematics 110 (Second ed.). Berlin, New York: Springer-Verlag. doi:10.1007/978-1-4612-0853-2. ISBN 978-0-387-94225-4. MR 1282723. • Postnikov, Mikhail Mikhaĭlovich (2004). Foundations of Galois Theory. With a foreword by P. J. Hilton. Reprint of the 1962 edition. Translated from the 1960 Russian original by Ann Swinfen. Dover Publications. ISBN 0-486-43518-0. MR 2043554. • Rotman, Joseph (1998). Galois Theory (Second ed.). Springer. doi:10.1007/978-1-4612-0617-0. ISBN 0387-98541-7. MR 1645586. • Völklein, Helmut (1996). Groups as Galois groups: an introduction. Cambridge Studies in Advanced Mathematics 53. Cambridge University Press. doi:10.1017/CBO9780511471117. ISBN 978-0-521-56280-5. MR 1405612. • van der Waerden, Bartel Leendert (1931). Moderne Algebra (in German). Berlin: Springer.. English translation (of 2nd revised edition): Modern algebra. New York: Frederick Ungar. 1949. (Later republished in English by Springer under the title “Algebra”.) • Pop, Florian (2001). "(Some) New Trends in Galois Theory and Arithmetic” (PDF).
Chapter 91
Galois module In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.
91.1 Examples • Given a field K, the multiplicative group (Ks )× of a separable closure of K is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of K (by Hilbert’s theorem 90, its first cohomology group is zero). • If X is a smooth proper scheme over a field K then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of K.
91.1.1
Ramification theory
Let K be a valued field (with valuation denoted v) and let L/K be a finite Galois extension with Galois group G. For an extension w of v to L, let Iw denote its inertia group. A Galois module ρ : G → Aut(V) is said to be unramified if ρ(Iw) = {1}.
91.2 Galois module structure of algebraic integers In classical algebraic number theory, let L be a Galois extension of a field K, and let G be the corresponding Galois group. Then the ring OL of algebraic integers of L can be considered as an OK[G]-module, and one can ask what its structure is. This is an arithmetic question, in that by the normal basis theorem one knows that L is a free K[G]module of rank 1. If the same is true for the integers, that is equivalent to the existence of a normal integral basis, i.e. of α in OL such that its conjugate elements under G give a free basis for OL over OK. This is an interesting question even (perhaps especially) when K is the rational number field Q. For example, if L = Q(√−3), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q(ζ) where ζ = exp(2πi/3). In fact all the subfields of the cyclotomic fields for p-th roots of unity for p a prime number have normal integral bases (over Z), as can be deduced from the theory of Gaussian periods (the Hilbert–Speiser theorem). On the other hand the Gaussian field does not. This is an example of a necessary condition found by Emmy Noether (perhaps known earlier?). What matters here is tame ramification. In terms of the discriminant D of L, and taking still K = Q, no prime p must divide D to the power p. Then Noether’s theorem states that tame ramification is necessary and 310
91.3. GALOIS REPRESENTATIONS IN NUMBER THEORY
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sufficient for OL to be a projective module over Z[G]. It is certainly therefore necessary for it to be a free module. It leaves the question of the gap between free and projective, for which a large theory has now been built up. A classical result, based on a result of David Hilbert, is that a tamely ramified abelian number field has a normal integral basis. This may be seen by using the Kronecker–Weber theorem to embed the abelian field into a cyclotomic field.[1]
91.3 Galois representations in number theory Many objects that arise in number theory are naturally Galois representations. For example, if L is a Galois extension of a number field K, the ring of integers OL of L is a Galois module over OK for the Galois group of L/K (see Hilbert– Speiser theorem). If K is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of K and its study leads to local class field theory. For global class field theory, the union of the idele class groups of all finite separable extensions of K is used instead. There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the ℓ-adic Tate modules of abelian varieties.
91.3.1
Artin representations
Let K be a number field. Emil Artin introduced a class of Galois representations of the absolute Galois group GK of K, now called Artin representations. These are the continuous finite-dimensional linear representations of GK on complex vector spaces. Artin’s study of these representations led him to formulate the Artin reciprocity law and conjecture what is now called the Artin conjecture concerning the holomorphy of Artin L-functions. Because of the incompatibility of the profinite topology on GK and the usual (Euclidean) topology on complex vector spaces, the image of an Artin representation is always finite.
91.3.2
ℓ-adic representations
Let ℓ be a prime number. An ℓ-adic representation of GK is a continuous group homomorphism ρ : GK → Aut(M) where M is either a finite-dimensional vector space over Qℓ (the algebraic closure of the ℓ-adic numbers Qℓ) or a finitely generated Zℓ-module (where Zℓ is the integral closure of Zℓ in Qℓ). The first examples to arise were the ℓ-adic cyclotomic character and the ℓ-adic Tate modules of abelian varieties over K. Other examples come from the Galois representations of modular forms and automorphic forms, and the Galois representations on ℓ-adic cohomology groups of algebraic varieties. Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of GQ under the ℓ-adic cyclotomic character is Z× ℓ . ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of Qℓ with C they can be identified with bona fide Artin representations.
91.3.3
Mod ℓ representations
These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic representation.
91.3.4
Local conditions on representations
There are numerous conditions on representations given by some property of the representation restricted to a decomposition group of some prime. The terminology for these conditions is somewhat chaotic, with different authors inventing different names for the same condition and using the same name with different meanings. Some of these conditions include: • Abelian representations. This means that the image of the Galois group in the representations is abelian. • Absolutely irreducible representations. These remain irreducible over an algebraic closure of the field.
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• Barsotti–Tate representations. These are similar to finite flat representations. • Crystalline representations. • de Rham representations. • Finite flat representations. (This name is a little misleading, as they are really profinite rather than finite.) These can be constructed as a projective limit of representations of the Galois group on a finite flat group scheme. • Good representations. These are similar to finite flat representations. • Hodge–Tate representations. • Irreducible representations. These are irreducible in the sense that the only subrepresentation is the whole space or zero. • Minimally ramified representations. • Modular representations. These are representations coming from a modular form. • Ordinary representations. These are 2-dimensional representations that are reducible with a 1-dimensional subrepresentation, such that the inertia group acts in a certain way on the submodule and the quotient. The exact condition depends on the author; for example it might act trivially on the quotient and by the character ε on the submodule. • Potentially something representations. This means that the representations restricted to an open subgroup of finite index has some property. • Reducible representations. These have a proper non-zero sub-representation. • Semistable representations. These are two dimensional representations related to the representations coming from semistable elliptic curves. • Tamely ramified representations. These are trivial on the (first) ramification group. • Unramified representations. These are trivial on the inertia group. • Wildly ramified representations. These are non-trivial on the (first) ramification group.
91.4 Representations of the Weil group If K is a local or global field, the theory of class formations attaches to K its Weil group WK, a continuous group homomorphism φ : WK → GK, and an isomorphism of topological groups
ab rK : CK →W ˜ K
where CK is K × or the idele class group IK/K × (depending on whether K is local or global) and W ab K is the abelianization of the Weil group of K. Via φ, any representation of GK can be considered as a representation of WK. However, WK can have strictly more representations than GK. For example, via rK the continuous complex characters of WK are in bijection with those of CK. Thus, the absolute value character on CK yields a character of WK whose image is infinite and therefore is not a character of GK (as all such have finite image). An ℓ-adic representation of WK is defined in the same way as for GK. These arise naturally from geometry: if X is a smooth projective variety over K, then the ℓ-adic cohomology of the geometric fibre of X is an ℓ-adic representation of GK which, via φ, induces an ℓ-adic representation of WK. If K is a local field of residue characteristic p ≠ ℓ, then it is simpler to study the so-called Weil–Deligne representations of WK.
91.5. SEE ALSO
91.4.1
313
Weil–Deligne representations
Let K be a local field. Let E be a field of characteristic zero. A Weil–Deligne representation over E of WK (or simply of K) is a pair (r, N) consisting of • a continuous group homomorphism r : WK → AutE(V), where V is a finite-dimensional vector space over E equipped with the discrete topology, • a nilpotent endomorphism N : V → V such that r(w)Nr(w)−1 = ||w||N for all w ∈ WK.[2] These representations are the same as the representations over E of the Weil–Deligne group of K. If the residue characteristic of K is different from ℓ, Grothendieck's ℓ-adic monodromy theorem sets up a bijection between ℓ-adic representations of WK (over Qℓ) and Weil–Deligne representations of WK over Qℓ (or equivalently over C). These latter have the nice feature that the continuity of r is only with respect to the discrete topology on V, thus making the situation more algebraic in flavor.
91.5 See also • Compatible system of ℓ-adic representations
91.6 Notes [1] Fröhlich (1983) p.8 [2] Here ||w|| is given by q v(w) K where qK is the size of the residue field of K and v(w) is such that w is equivalent to the −v(w)th power of the (arithmetic) Frobenius of WK.
91.7 References • Kudla, Stephen S. (1994), “The local Langlands correspondence: the non-archimedean case”, Motives, Part 2, Proc. Sympos. Pure Math. 55, Providence, R.I.: Amer. Math. Soc., pp. 365–392, ISBN 978-0-8218-1635-6 • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, Zbl 0948.11001, MR 1737196 • Tate, John (1979), “Number theoretic background”, Automorphic forms, representations, and L-functions, Part 2, Proc. Sympos. Pure Math. 33, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 978-0-8218-1437-6
91.8 Further reading • Snaith, Victor P. (1994), Galois module structure, Fields Institute monographs, Providence, RI: American Mathematical Society, ISBN 0-8218-0264-X, Zbl 0830.11042 • Fröhlich, Albrecht (1983), Galois module structure of algebraic integers, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 1, Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, ISBN 3-540-11920-5, Zbl 0501.12012
Chapter 92
Generic polynomial In Galois theory, a branch of modern algebra, a generic polynomial for a finite group G and field F is a monic polynomial P with coefficients in the field L = F(t 1 , ..., tn) of F with n indeterminates adjoined, such that the splitting field M of P has Galois group G over L, and such that every extension K/F with Galois group G can be obtained as the splitting field of a polynomial which is the specialization of P resulting from setting the n indeterminates to n elements of F. This is sometimes called F-generic relative to the field F, with a Q-generic polynomial, generic relative to the rational numbers, being called simply generic. The existence, and especially the construction, of a generic polynomial for a given Galois group provides a complete solution to the inverse Galois problem for that group. However, not all Galois groups have generic polynomials, a counterexample being the cyclic group of order eight.
92.1 Groups with generic polynomials • The symmetric group Sn. This is trivial, as xn + t1 xn−1 + · · · + tn is a generic polynomial for Sn. • Cyclic groups Cn, where n is not divisible by eight. Lenstra showed that a cyclic group does not have a generic polynomial if n is divisible by eight, and Smith explicitly constructs such a polynomial in case n is not divisible by eight. • The cyclic group construction leads to other classes of generic polynomials; in particular the dihedral group Dn has a generic polynomial if and only if n is not divisible by eight. • The quaternion group Q8 . • Heisenberg groups Hp3 for any odd prime p. • The alternating group A4 . • The alternating group A5 . • Reflection groups defined over Q, including in particular groups of the root systems for E 6 , E 7 , and E 8 . • Any group which is a direct product of two groups both of which have generic polynomials. • Any group which is a wreath product of two groups both of which have generic polynomials. 314
92.2. EXAMPLES OF GENERIC POLYNOMIALS
315
92.2 Examples of generic polynomials Generic polynomials are known for all transitive groups of degree 5 or less.
92.3 Generic Dimension The generic dimension for a finite group G over a field F, denoted gdF G , is defined as the minimal number of parameters in a generic polynomial for G over F, or ∞ if no generic polynomial exists. Examples: • gdQ A3 = 1 • gdQ S3 = 1 • gdQ D4 = 2 • gdQ S4 = 2 • gdQ D5 = 2 • gdQ S5 = 2
92.4 Publications • Jensen, Christian U., Ledet, Arne, and Yui, Noriko, Generic Polynomials, Cambridge University Press, 2002
Chapter 93
Genus character In number theory, a genus character of a quadratic number field K is a character of the genus group of K. In other words, it is a quadratic character (also known as a real character) of the narrow class group of K. Reinterpreting this using the Artin map, the collection of genus characters can also be thought of as the unramified quadratic characters of the absolute Galois group of K (i.e. the characters that factor through the Galois group of the genus field of K).
93.1 References • Bertolini, Massimo; Darmon, Henri (2009), “The rationality of Stark-Heegner points over genus fields of real quadratic fields”, Annals of Mathematics 170: 343–369, doi:10.4007/annals.2009.170.343, ISSN 0003-486X, MR 2521118 • Section 12.5 of Iwaniec, Henryk, Topics in classical automorphic forms • Section 2.3 of Lemmermeyer, Franz, Reciprocity laws: From Euler to Eisenstein
316
Chapter 94
Genus field In algebraic number theory, the genus field G of an algebraic number field K is the maximal abelian extension of K which is obtained by composing an absolutely abelian field with K and which is unramified at all finite primes of K. The genus number of K is the degree [G:K] and the genus group is the Galois group of G over K. If K is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of K unramified at all finite primes: this definition was used by Leopoldt and Hasse. If K=Q(√m) (m squarefree) is a quadratic field of discriminant D, the genus field of K is a composite of quadratic fields. Let pi run over the prime factors of D. For each such prime p, define p∗ as follows:
p∗ = ±p ≡ 1
(mod 4) if podd is ;
2∗ = −4, 8, −8 as according m ≡ 3
(mod 4), 2 (mod 8), −2
(mod 8).
Then the genus field is the composite K(√pᵢ∗ ).
94.1 See also • Hilbert class field
94.2 References • Ishida, Makoto (1976). The genus fields of algebraic number fields. Lecture Notes in Mathematics 555. Springer-Verlag. ISBN 3-540-08000-7. Zbl 0353.12001. • Janusz, Gerald (1973). Algebraic Number Fields. Pure and Applied Mathematics 55. Academic Press. ISBN 0-12-380250-4. Zbl 0307.12001. • Lemmermeyer, Franz (2000). Reciprocity laws. From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer-Verlag. ISBN 3-540-66957-4. MR 1761696. Zbl 0949.11002.
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Chapter 95
Global field In mathematics, the term global field refers to a field that is either: • an algebraic number field, i.e., a finite extension of Q, or • a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq(T), the field of rational functions in one variable over the finite field with q elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.[1]
95.1 Formal definitions Main articles: Algebraic number field and Function field of an algebraic variety A global field is one of the following: An algebraic number field An algebraic number field F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. A function field of an algebraic variety of an algebraic curve over a finite field
A function field of a variety is the set of all rational functions on that variety. On an algebraic curve (i.e. a onedimensional variety V) over a finite field, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data which agree on the intersections of open affines. This technically defines the rational functions on V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.
95.2 Analogies between the two classes of fields There are a number of formal similarities between the two kinds of fields. A field of either type has the property that all of its completions are locally compact fields (see local fields). Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index. In each case, one has the product formula for non-zero elements x: ∏
|x|v = 1.
v
318
95.3. THEOREMS
319
The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory. The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekind and Heinrich M. Weber in the nineteenth century. The more strict analogy expressed by the 'global field' idea, in which a Riemann surface’s aspect as algebraic curve is mapped to curves defined over a finite field, was built up during the 1930s, culminating in the Riemann hypothesis for curves over finite fields settled by André Weil in 1940. The terminology may be due to Weil, who wrote his Basic Number Theory (1967) in part to work out the parallelism. It is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example. The analogy was also influential in the development of Iwasawa theory and the Main Conjecture. The proof of the fundamental lemma in the Langlands program also made use of techniques that reduced the number field case to the function field case.
95.3 Theorems 95.3.1
Hasse-Minkowski theorem
Main article: Hasse–Minkowski theorem The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a global field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field.
95.3.2
Artin reciprocity law
Main article: Artin reciprocity law Artin’s reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the Hasse local–global principle. It can be described in terms of cohomology as follows: Let Lv⁄Kv be a Galois extension of local fields with Galois group G. The local reciprocity law describes a canonical isomorphism
ab θv : Kv× /NLv /Kv (L× v)→G ,
called the local Artin symbol, the local reciprocity map or the norm reside symbol.[2][3] Let L⁄K be a Galois extension of global fields and CL stand for the idèle class group of L. The maps θv for different places v of K can be assembled into a single global symbol map by multiplying the local components of an idèle class. One of the statements of the Artin reciprocity law is that this results in the canonical isomorphism[4][5]
95.4 Notes [1] Artin & Whaples 1945 and Artin & Whaples 1946 [2] Serre (1967) p.140 [3] Serre (1979) p.197 [4] Neukirch (1999) p.391 [5] Jürgen Neukirch, Algebraische Zahlentheorie, Springer, 1992, p. 408. In fact, a more precise version of the reciprocity law keeps track of the ramification.
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95.5 References • Artin, Emil; Whaples, George (1945), “Axiomatic characterization of fields by the product formula for valuations”, Bull. Amer. Math. Soc. 51: 469–492, doi:10.1090/S0002-9904-1945-08383-9, MR 0013145 • Artin, Emil; Whaples, George (1946), “A note on axiomatic characterization of fields”, Bull. Amer. Math. Soc. 52: 245–247, doi:10.1090/S0002-9904-1946-08549-3, MR 0015382 • J.W.S. Cassels, “Global fields”, in J.W.S. Cassels and A. Frohlich (eds), Algebraic number theory, Academic Press, 1973. Chap.II, pp. 45–84. • J.W.S. Cassels, “Local fields”, Cambridge University Press, 1986, ISBN 0-521-31525-5. P.56. • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, MR 2392026, Zbl 1136.11001
Chapter 96
Glossary of field theory Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
96.1 Definition of a field A field is a commutative ring (F,+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division. The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted by F × ; The ring of polynomials in the variable x with coefficients in F is denoted by F[x].
96.2 Basic definitions Characteristic The characteristic of the field F is the smallest positive integer n such that n·1 = 0; here n·1 stands for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbers have characteristic 0, while the finite field Zp has characteristic p. Subfield A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field. Prime field The prime field of the field F is the unique smallest subfield of F. Extension field If F is a subfield of E then E is an extension field of F. We then also say that E/F is a field extension. Degree of an extension Given an extension E/F, the field E can be considered as a vector space over the field F, and the dimension of this vector space is the degree of the extension, denoted by [E : F]. Finite extension A finite extension is a field extension whose degree is finite. Algebraic extension If an element α of an extension field E over F is the root of a non-zero polynomial in F[x], then α is algebraic over F. If every element of E is algebraic over F, then E/F is an algebraic extension. Generating set Given a field extension E/F and a subset S of E, we write F(S) for the smallest subfield of E that contains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations +,−,*,/ on the elements of F and S. If E = F(S) we say that E is generated by S over F. Primitive element An element α of an extension field E over a field F is called a primitive element if E=F(α), the smallest extension field containing α. Such an extension is called a simple extension. 321
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Splitting field A field extension generated by the complete factorisation of a polynomial. Normal extension A field extension generated by the complete factorisation of a set of polynomials. Separable extension An extension generated by roots of separable polynomials. Perfect field A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect. Imperfect degree Let F be a field of characteristic p>0; then F p is a subfield. The degree [F:F p ] is called the imperfect degree of F. The field F is perfect if and only if its imperfect degree is 1. For example, if F is a function field of n variables over a finite field of characteristic p>0, then its imperfect degree is pn .[1] Algebraically closed field A field F is algebraically closed if every polynomial in F[x] has a root in F; equivalently: every polynomial in F[x] is a product of linear factors. Algebraic closure An algebraic closure of a field F is an algebraic extension of F which is algebraically closed. Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F. Transcendental Those elements of an extension field of F that are not algebraic over F are transcendental over F. Algebraically independent elements Elements of an extension field of F are algebraically independent over F if they don't satisfy any non-zero polynomial equation with coefficients in F. Transcendence degree The number of algebraically independent transcendental elements in a field extension. It is used to define the dimension of an algebraic variety.
96.3 Homomorphisms Field homomorphism A field homomorphism between two fields E and F is a function f: E→F such that f(x + y) = f(x) + f(y) and f(xy) = f(x) f(y) for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x−1 ) = f(x)−1 for x in E with x ≠ 0, and that f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are called isomorphic if there exists a bijective homomorphism f : E → F. The two fields are then identical for all practical purposes; however, not necessarily in a unique way. See, for example, complex conjugation.
96.4 Types of fields Finite field A field with finitely many elements. Ordered field A field with a total order compatible with its operations.
96.5. FIELD EXTENSIONS
323
Rational numbers Real numbers Complex numbers Number field Finite extension of the field of rational numbers. Algebraic numbers The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in algebraic number theory. Quadratic field A degree-two extension of the rational numbers. Cyclotomic field An extension of the rational numbers generated by a root of unity. Totally real field A number field generated by a root of a polynomial, having all its roots real numbers. Formally real field Real closed field Global field A number field or a function field of one variable over a finite field. Local field A completion of some global field (w.r.t. a prime of the integer ring). Complete field A field complete w.r.t. to some valuation. Pseudo algebraically closed field A field in which every variety has a rational point.[2] Henselian field A field satisfying Hensel lemma w.r.t. some valuation. A generalization of complete fields. Hilbertian field A field satisfying Hilbert’s irreducibility theorem: formally, one for which the projective line is not thin in the sense of Serre.[3][4] Kroneckerian field A totally real algebraic number field or a totally imaginary quadratic extension of a totally real field.[5] CM-field or J-field An algebraic number field which is a totally imaginary quadratic extension of a totally real field.[6] Linked field A field over which no biquaternion algebra is a division algebra.[7] Frobenius field A pseudo algebraically closed field whose absolute Galois group has the embedding property.[8]
96.5 Field extensions Let E / F be a field extension. Algebraic extension An extension in which every element of E is algebraic over F. Simple extension An extension which is generated by a single element, called a primitive element, or generating element.[9] The primitive element theorem classifies such extensions.[10]
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Normal extension An extension that splits a family of polynomials: every root of the minimal polynomial of an element of E over F is also in E. Separable extension An algebraic extension in which the minimal polynomial of every element of E over F is a separable polynomial, that is, has distinct roots.[11] Galois extension A normal, separable field extension. Primary extension An extension E/F such that the algebraic closure of F in E is purely inseparable over F; equivalently, E is linearly disjoint from the separable closure of F.[12] Purely transcendental extension An extension E/F in which every element of E not in F is transcendental over F.[13][14] Regular extension An extension E/F such that E is separable over F and F is algebraically closed in E.[12] Simple radical extension A simple extension E/F generated by a single element α satisfying αn = b for an element b of F. In characteristic p, we also take an extension by a root of an Artin–Schreier polynomial to be a simple radical extension.[15] Radical extension A tower F = F0 < F1 < · · · < Fk = E where each extension Fi /Fi−1 is a simple radical extension.[15] Self-regular extension An extension E/F such that E⊗FE is an integral domain.[16] Totally transcendental extension An extension E/F such that F is algebraically closed in F.[14] Distinguished class A class C of field extensions with the three properties[17] 1. If E is a C-extension of F and F is a C-extension of K then E is a C-extension of K. 2. If E and F are C-extensions of K in a common overfield M, then the compositum EF is a C-extension of K. 3. If E is a C-extension of F and E>K>F then E is a C-extension of K.
96.6 Galois theory Galois extension A normal, separable field extension. Galois group The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups. Kummer theory The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theory of quadratic extensions. Artin–Schreier theory Covers an exceptional case of Kummer theory, in characteristic p. Normal basis A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively. Tensor product of fields A different foundational piece of algebra, including the compositum operation (join of fields).
96.7. EXTENSIONS OF GALOIS THEORY
325
96.7 Extensions of Galois theory Inverse problem of Galois theory Given a group G, find an extension of the rational number or other field with G as Galois group. Differential Galois theory The subject in which symmetry groups of differential equations are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie founded the theory of Lie groups. It has not, probably, reached definitive form. Grothendieck’s Galois theory A very abstract approach from algebraic geometry, introduced to study the analogue of the fundamental group.
96.8 References [1] Fried & Jarden (2008) p.45 [2] Fried & Jarden (2008) p.214 [3] Serre (1992) p.19 [4] Schinzel (2000) p.298 [5] Schinzel (2000) p.5 [6] Washington, Lawrence C. (1996). Introduction to Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-38794762-0. Zbl 0966.11047. [7] Lam (2005) p.342 [8] Fried & Jarden (2008) p.564 [9] Roman (2007) p.46 [10] Lang (2002) p.243 [11] Fried & Jarden (2008) p.28 [12] Fried & Jarden (2008) p.44 [13] Roman (2007) p.102 [14] Isaacs, I. Martin (1994). Algebra: A Graduate Course. Graduate studies in mathematics 100. American Mathematical Society. p. 389. ISBN 0-8218-4799-6. ISSN 1065-7339. [15] Roman (2007) p.273 [16] Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. p. 427. ISBN 1-85233-587-4. Zbl 1003.00001. [17] Lang (2002) p.228
• Adamson, Iain T. (1982). Introduction to Field Theory (2nd ed.). Cambridge University Press. ISBN 0-52128658-1. • Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051. • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: SpringerVerlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556
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• Roman, Steven (2007). Field Theory. Graduate Texts in Mathematics 158. Springer-Verlag. ISBN 0-38727678-5. • Serre, Jean-Pierre (1989). Lectures on the Mordell-Weil Theorem. Aspects of Mathematics E15. Translated and edited by Martin Brown from notes by Michel Waldschmidt. Braunschweig etc.: Friedr. Vieweg & Sohn. Zbl 0676.14005. • Serre, Jean-Pierre (1992). Topics in Galois Theory. Research Notes in Mathematics 1. Jones and Bartlett. ISBN 0-86720-210-6. Zbl 0746.12001. • Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001.
Chapter 97
Golod–Shafarevich theorem In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite.
97.1 The inequality Let A = K be the free algebra over a field K in n = d + 1 non-commuting variables xi. Let J be the 2-sided ideal of A generated by homogeneous elements fj of A of degree dj with 2 ≤ d1 ≤ d2 ≤ ... where dj tends to infinity. Let ri be the number of dj equal to i. Let B=A/J, a graded algebra. Let bj = dim Bj. The fundamental inequality of Golod and Shafarevich states that
bj ≥ nbj−1 −
j ∑
bj−i ri .
i=2
As a consequence: • B is infinite-dimensional if ri ≤ d2 /4 for all i • if B is finite-dimensional, then ri > d2 /4 for some i.
97.2 Applications This result has important applications in combinatorial group theory: • If G is a nontrivial finite p-group, then r > d2 /4 where d = dim H 1 (G,Z/pZ) and r = dim H 2 (G,Z/pZ) (the mod p cohomology groups of G). In particular if G is a finite p-group with minimal number of generators d and has r relators in a given presentation, then r > d2 /4. • For each prime p, there is an infinite group G generated by three elements in which each element has order a power of p. The group G provides a counterexample to the generalised Burnside conjecture: it is a finitely generated infinite torsion group, although there is no uniform bound on the order of its elements. 327
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CHAPTER 97. GOLOD–SHAFAREVICH THEOREM
In class field theory, the class field tower of a number field K is created by iterating the Hilbert class field construction. The class field tower problem asks whether this tower is always finite; Hasse (1926) attributed this question to Furtwangler, though Furtwangler said he had heard it from Schreier. Another consequence of the Golod–Shafarevich theorem is that such towers may be infinite (in other words, do not always terminate in a field equal to its Hilbert class field). Specifically, • Let K be an imaginary quadratic field whose discriminant has at least 6 prime factors. Then the maximal unramified 2-extension of K has infinite degree. More generally, a number field with sufficiently many prime factors in the discriminant has an infinite class field tower.
97.3 References • Golod, E.S; Shafarevich, I.R. (1964), “On the class field tower”, Izv. Akad. Nauk SSSSR 28: 261–272 (in Russian) MR 0161852 • Golod, E.S (1964), “On nil-algebras and finitely approximable p-groups.”, Izv. Akad. Nauk SSSSR 28: 273–276 (in Russian) MR 0161878 • Herstein, I.N. (1968). Noncommutative rings. Carus Mathematical Monographs. MAA. ISBN 0-88385-039-7. See Chapter 8. • Johnson, D.L. (1980). “Topics in the Theory of Group Presentations” (1st ed.). Cambridge University Press. ISBN 0-521-23108-6. See chapter VI. • Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). SpringerVerlag. p. 180. ISBN 3-540-63003-1. Zbl 0819.11044. • Narkiewicz, Władysław (2004). Elementary and analytic theory of algebraic numbers. Springer Monographs in Mathematics (3rd ed.). Berlin: Springer-Verlag. p. 194. ISBN 3-540-21902-1. Zbl 1159.11039. • Roquette, Peter (1986). “On class field towers”. In Cassels, J. W. S.; Fröhlich, A.. Algebraic number theory, Proceedings of the instructional conference held at the University of Sussex, Brighton, September 1–17, 1965 (Reprint of the 1967 original ed.). London: Academic Press. pp. 231–249. ISBN 0-12-163251-2. • Serre, J.-P. (2002), “Galois Cohomology,” Springer-Verlag. ISBN 3-540-42192-0. See Appendix 2. (Translation of Cohomologie Galoisienne, Lecture Notes in Mathematics 5, 1973.)
Chapter 98
Grothendieck’s Galois theory In mathematics, Grothendieck’s Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s. The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the ˆ , which is the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted Z inverse limit of the cyclic additive groups Z/nZ — or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group, so that it is specified by giving some permutation of X. ˆ as the profinite Galois In the above example, a connection with classical Galois theory can be seen by regarding Z group Gal(F/F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the zn map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n.Z of the fundamental group of the punctured disk. The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor Φ, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type G ≅ Aut(Φ), the latter being the group of automorphisms (self-natural equivalences) of Φ. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite. To see how this applies to the case of fields, one has to study the tensor product of fields. Later developments in topos theory make this all part of a theory of atomic toposes.
98.1 References • Grothendieck, A. et al. (1971). SGA1 Revêtements étales et groupe fondamental, 1960–1961'. Lecture Notes in Mathematics 224. Springer Verlag. • Joyal, André; Tierney, Myles (1984). An Extension of the Galois Theory of Grothendieck. Memoirs of the American Mathematical Society. Proquest Info & Learning. ISBN 0-8218-2312-4. • Borceux, F. and Janelidze, G., Cambridge University Press (2001). Galois theories, ISBN 0-521-80309-8 (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.) • Szamuely, T., Galois Groups and Fundamental Groups, Cambridge University Press, 2009. 329
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• Dubuc, E. J and de la Vega, C. S., On the Galois theory of Grothendieck, http://arxiv.org/abs/math/0009145v1
Chapter 99
Ground field In mathematics, a ground field is a field K fixed at the beginning of the discussion.
99.1 Use It is used in various areas of algebra:
99.2 In linear algebra In linear algebra, the concept of a vector space may be developed over any field.
99.2.1
In algebraic geometry
In algebraic geometry, in the foundational developments of André Weil the use of fields other than the complex numbers was essential to expand the definitions to include the idea of abstract algebraic variety over K, and generic point relative to K.[1]
99.2.2
In Lie theory
Reference to a ground field may be common in the theory of Lie algebras (qua vector spaces) and algebraic groups (qua algebraic varieties).
99.2.3
In Galois theory
In Galois theory, given a field extension L/K, the field K that is being extended may be considered the ground field for an argument or discussion. Within algebraic geometry, from the point of view of scheme theory, the spectrum Spec(K) of the ground field K plays the role of final object in the category of K-schemes, and its structure and symmetry may be richer than the fact that the space of the scheme is a point might suggest.
99.2.4
In Diophantine geometry
In diophantine geometry the characteristic problems of the subject are those caused by the fact that the ground field K is not taken to be algebraically closed. The field of definition of a variety given abstractly may be smaller than the ground field, and two varieties may become isomorphic when the ground field is enlarged, a major topic in Galois cohomology.[2] 331
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99.3 Notes [1] Hazewinkel, Michiel, ed. (2001), “Abstract algebraic geometry”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4 [2] Hazewinkel, Michiel, ed. (2001), “Form of an algebraic group”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4
Chapter 100
Group cohomology This article is about homology and cohomology of a group. For homology or cohomology groups of a space or other object, see Homology (mathematics). In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of Gn representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups H n (G, M ) . The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology. The techniques of group cohomology can also be extended to the case that instead of a G-module, G acts on a nonabelian G-group; in effect, a generalization of a module to non-Abelian coefficients. These algebraic ideas are closely related to topological ideas. The group cohomology of a group G can be thought of as, and is motivated by, the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of Z can be thought of as the singular cohomology of the circle S1 , and similarly for Z/2Z and P∞ (R). A great deal is known about the cohomology of groups, including interpretations of low-dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.
100.1 Motivation A general paradigm in group theory is that a group G should be studied via its group representations. A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M. We will write G multiplicatively and M additively. Given such a G-module M, it is natural to consider the submodule of G-invariant elements:
M G = {x ∈ M | ∀g ∈ G : gx = x}. Now, if N is a submodule of M (i.e. a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in M/N are found as the quotient of the invariants in M by those in N: being invariant 'modulo N ' is broader. The first group cohomology H 1 (G,N) precisely measures the difference. The group cohomology functors H* in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence. 333
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100.2 Formal constructions In this article, G is a finite group. The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property f(gx) = g(f(x)) for all g in G and x in M). This category of G-modules is an abelian category with enough injectives (since it is isomorphic to the category of all modules over the group ring Z[G]). Sending each module M to the group of invariants MG yields a functor from this category to the category Ab of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors; their values are abelian groups and they are denoted by Hn (G, M), “the n-th cohomology group of G with coefficients in M". H 0 (G, M) is identified with MG .
100.2.1
Long exact sequence of cohomology
In practice, one often computes the cohomology groups using the following fact: if
0→L→M →N →0 is a short exact sequence of G-modules, then a long exact sequence δ0
δ1
0 → LG → M G → N G →H 1 (G, L) → H 1 (G, M ) → H 1 (G, N )→H 2 (G, L) → · · · is induced. The maps δn are called the "connecting homomorphisms" and can be obtained from the snake lemma.[1]
100.2.2
Cochain complexes
Rather than using the machinery of derived functors, the cohomology groups can also be defined more concretely, as follows.[2] For n ≥ 0, let Cn (G, M) be the group of all functions from Gn to M. This is an abelian group; its elements are called the (inhomogeneous) n-cochains. The coboundary homomorphisms
dn : C n (G, M ) → C n+1 (G, M ) are defined as (dn φ) (g1 , . . . , gn+1 ) = g1 · φ(g2 , . . . , gn+1 ) +
n ∑ (−1)i φ(g1 , . . . , gi−1 , gi gi+1 , gi+2 , . . . , gn+1 ) i=1
+ (−1)n+1 φ(g1 , . . . , gn ) The crucial thing to check here is
dn+1 ◦ dn = 0 thus we have a cochain complex and we can compute cohomology. For n ≥ 0, define the group of n-cocycles as:
Z n (G, M ) = ker(dn ) and the group of n-coboundaries as {
B 0 (G, M ) = 0 B n (G, M ) = im(dn−1 ), n ≥ 1
100.2. FORMAL CONSTRUCTIONS
335
and
H n (G, M ) = Z n (G, M )/B n (G, M ).
100.2.3
The functors Extn and formal definition of group cohomology
Yet another approach is to treat G-modules as modules over the group ring Z[G], which allows one to define group cohomology via Ext functors:
H n (G, M ) = ExtnZ[G] (Z, M ), where M is a Z[G]-module. Here Z is treated as the trivial G-module: every element of G acts as the identity. These Ext groups can also be computed via a projective resolution of Z, the advantage being that such a resolution only depends on G and not on M. We recall the definition of Ext more explicitly for this context. Let F be a projective Z[G]-resolution (e.g. a free Z[G]-resolution) of the trivial Z[G]-module Z:
· · · → Fn → Fn−1 → · · · → F0 → Z. e.g., one may always take the resolution of group rings, Fn = Z[Gn+1 ], with morphisms
fn : Z[Gn+1 ] → Z[Gn ],
(g0 , g1 , . . . , gn ) 7→
n ∑ (−1)i (g0 , . . . , gbi , . . . , gn ). i=0
Recall that for Z[G]-modules N and M, HomG(N, M) is an abelian group consisting of Z[G]-homomorphisms from N to M. Since HomG(–, M) is a contravariant functor and reverses the arrows, applying HomG(–, M) to F termwise produces a cochain complex HomG(F, M):
· · · ← HomG (Fn , M ) ← HomG (Fn−1 , M ) ← · · · ← HomG (F0 , M ) ← HomG (Z, M ). The cohomology groups H*(G, M) of G with coefficients in the module M are defined as the cohomology of the above cochain complex:
H n (G, M ) = H n (HomG (F, M )) for n ≥ 0. This construction initially leads to a coboundary operator that acts on the “homogeneous” cochains. These are the elements of HomG(F, M) i.e. functions φn: Gn → M that obey
gϕn (g1 , g2 , . . . , gn ) = ϕn (gg1 , gg2 , . . . , ggn ). The coboundary operator δ: Cn → C n+1 is now naturally defined by, for example,
δϕ2 (g1 , g2 , g3 ) = ϕ2 (g2 , g3 ) − ϕ2 (g1 , g3 ) + ϕ2 (g1 , g2 ). The relation to the coboundary operator d that was defined in the previous section, and which acts on the “inhomogeneous” cochains φ , is given by reparameterizing so that
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φ2 (g1 , g2 ) = ϕ3 (1, g1 , g1 g2 ) φ3 (g1 , g2 , g3 ) = ϕ4 (1, g1 , g1 g2 , g1 g2 g3 ), and so on. Thus
dφ2 (g1 , g2 , g3 ) = δϕ3 (1, g1 , g1 g2 , g1 g2 g3 ) = ϕ3 (g1 , g1 g2 , g1 g2 g3 ) − ϕ3 (1, g1 g2 , g1 g2 g3 ) + ϕ3 (1, g1 , g1 g2 g3 ) − ϕ3 (1, g1 , g1 g2 ) = g1 ϕ3 (1, g2 , g2 g3 ) − ϕ3 (1, g1 g2 , g1 g2 g3 ) + ϕ3 (1, g1 , g1 g2 g3 ) − ϕ3 (1, g1 , g1 g2 ) = g1 φ2 (g2 , g3 ) − φ2 (g1 g2 , g3 ) + φ2 (g1 , g2 g3 ) − φ2 (g1 , g2 ), as in the preceding section.
100.2.4
Group homology
Dually to the construction of group cohomology there is the following definition of group homology: given a Gmodule M, set DM to be the submodule generated by elements of the form g·m − m, g ∈ G, m ∈ M. Assigning to M its so-called coinvariants, the quotient
MG := M /DM, is a right exact functor. Its left derived functors are by definition the group homology
Hn (G, M ) Note that the superscript/subscript convention for cohomology/homology agrees with the convention for group invariants/coinvariants, while which is denoted “co-" switches: • superscripts correspond to cohomology H* and invariants XG while • subscripts correspond to homology H∗ and coinvariants XG := X/G. The covariant functor which assigns MG to M is isomorphic to the functor which sends M to Z ⊗Z[G] M, where Z is endowed with the trivial G-action. Hence one also gets an expression for group homology in terms of the Tor functors,
Hn (G, M ) = TornZ[G] (Z, M ) Recall that the tensor product N ⊗Z[G] M is defined whenever N is a right Z[G]-module and M is a left Z[G]module. If N is a left Z[G]-module, we turn it into a right Z[G]-module by setting a g = g−1 a for every g ∈ G and every a ∈ N. This convention allows to define the tensor product N ⊗Z[G] M in the case where both M and N are left Z[G]-modules. Specifically, the homology groups Hn(G, M) can be computed as follows. Start with a projective resolution F of the trivial Z[G]-module Z, as in the previous section. Apply the covariant functor ⋅ ⊗Z[G] M to F termwise to get a chain complex F ⊗Z[G] M:
· · · → Fn ⊗Z[G] M → Fn−1 ⊗Z[G] M → · · · → F0 ⊗Z[G] M → Z ⊗Z[G] M. Then Hn(G, M) are the homology groups of this chain complex, Hn (G, M ) = Hn (F ⊗Z[G] M ) for n ≥ 0. Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups.
100.3. FUNCTORIAL MAPS IN TERMS OF COCHAINS
337
100.3 Functorial maps in terms of cochains 100.3.1
Connecting homomorphisms
For a short exact sequence 0 → L → M → N → 0, the connecting homomorphisms δn : H n (G, N) → H n+1 (G, L) can be described in terms of inhomogeneous cochains as follows.[3] If c is an element of H n (G, N) represented by an n-cocycle φ : Gn → N, then δn (c) is represented by dn (ψ), where ψ is an n-cochain Gn → M “lifting” φ (i.e. such that φ is the composition of ψ with the surjective map M → N).
100.4 Non-abelian group cohomology See also: Nonabelian cohomology Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G. The zeroth cohomology of G with coefficients in A is defined to be the subgroup
H 0 (G, A) = AG , of elements of A fixed by G. The first cohomology of G with coefficients in A is defined as 1-cocycles modulo an equivalence relation instead of by 1-coboundaries. The condition for a map φ to be a 1-cocycle is that φ(gh) = φ(g)[gφ(h)] and φ ∼ φ′ if there is an a in A such that aφ′ (g) = φ(g) · (ga) . In general, H 1 (G, A) is not a group when A is non-abelian. It instead has the structure of a pointed set – exactly the same situation arises in the 0th homotopy group, π0 (X; x) which for a general topological space is not a group but a pointed set. Note that a group is in particular a pointed set, with the identity element as distinguished point. Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let
1→A→B→C→1 be a short exact sequence of G-groups, then there is an exact sequence of pointed sets
1 → AG → B G → C G → H 1 (G, A) → H 1 (G, B) → H 1 (G, C).
100.5 Connections with topological cohomology theories Group cohomology can be related to topological cohomology theories: to the topological group G there is an associated classifying space BG. (If G has no topology about which we care, then we assign the discrete topology to G. In this case, BG is an Eilenberg-MacLane space K(G,1), whose fundamental group is G and whose higher homotopy groups vanish). The n-th cohomology of BG, with coefficients in M (in the topological sense), is the same as the group cohomology of G with coefficients in M. This will involve a local coefficient system unless M is a trivial G-module. The connection holds because the total space EG is contractible, so its chain complex forms a projective resolution of M. These connections are explained in (Adem & Milgram 2004), Chapter II. When M is a ring with trivial G-action, we inherit good properties which are familiar from the topological context: in particular, there is a cup product under which
H ∗ (G; M ) =
⊕ n
H n (G; M )
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is a graded module, and a Künneth formula applies. If, furthermore, M = k is a field, then H*(G; k) is a graded k-algebra. In this case, the Künneth formula yields
H ∗ (G1 × G2 ; k) ∼ = H ∗ (G1 ; k) ⊗ H ∗ (G2 ; k). For example, let G be the group with two elements, under the discrete topology. The real projective space P∞ (R) is a classifying space for G. Let k = F2 , the field of two elements. Then H ∗ (G; k) ∼ = k[x], a polynomial k-algebra on a single generator, since this is the cellular cohomology ring of P∞ (R). Hence, as a second example, if G is an elementary abelian 2-group of rank r, and k = F2 , then the Künneth formula gives
H ∗ (G; k) ∼ = k[x1 , . . . , xr ] a polynomial k-algebra generated by r classes in H 1 (G; k).
100.6 Properties In the following, let M be a G-module.
100.6.1
Functoriality
Group cohomology depends contravariantly on the group G, in the following sense: if f : H → G is a group homomorphism, then we have a naturally induced morphism Hn (G,M) → Hn (H,M) (where in the latter, M is treated as an H-module via f). Given a morphism of G-modules M→N, one gets a morphism of cohomology groups in the Hn (G,M) → Hn (G,N).
100.6.2 H 1 The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) f : G → M satisfying f(ab) = f(a) + af(b) for all a, b in G, modulo the so-called principal crossed homomorphisms, i.e. maps f : G → M given by f(a) = am−m for some fixed m ∈ M. This follows from the definition of cochains above. If the action of G on M is trivial, then the above boils down to H 1 (G,M) = Hom(G, M), the group of group homomorphisms G → M.
100.6.3 H 2 If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H 2 (G,M) is in oneto-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H 2 (G,M) classifies the isomorphism classes of all extensions of G by M in which the induced action of G on M by inner automorphisms agrees with the given action.
100.6.4
Change of group
The Hochschild–Serre spectral sequence relates the cohomology of a normal subgroup N of G and the quotient G/N to the cohomology of the group G (for (pro-)finite groups G). From it, one gets the inflation-restriction exact sequence.
100.7. HISTORY AND RELATION TO OTHER FIELDS
100.6.5
339
Cohomology of finite groups is torsion
The cohomology groups of finite groups are all torsion. Indeed, by Maschke’s theorem the category of representations of a finite group is semi-simple over any field of characteristic zero (or more generally, any field whose characteristic does not divide the order of the group), hence, viewing group cohomology as a derived functor in this abelian category, one obtains that it is zero. The other argument is that over a field of characteristic zero, the group algebra of a finite group is a direct sum of matrix algebras (possibly over division algebras which are extensions of the original field), while a matrix algebra is Morita equivalent to its base field and hence has trivial cohomology.
100.7 History and relation to other fields The low-dimensional cohomology of a group was classically studied in other guises, long before the notion of group cohomology was formulated in 1943–45. The first theorem of the subject can be identified as Hilbert’s Theorem 90 in 1897; this was recast into Noether's equations in Galois theory (an appearance of cocycles for H 1 ). The idea of factor sets for the extension problem for groups (connected with H 2 ) arose in the work of Hölder (1893), in Issai Schur's 1904 study of projective representations, in Schreier's 1926 treatment, and in Richard Brauer's 1928 study of simple algebras and the Brauer group. A fuller discussion of this history may be found in (Weibel 1999, pp. 806–811). In 1941, while studying H 2 (G, Z) (which plays a special role in groups), Hopf discovered what is now called Hopf’s integral homology formula (Hopf 1942), which is identical to Schur’s formula for the Schur multiplier of a finite, finitely presented group: H2 (G, Z) ∼ = (R ∩ [F, F ])/[F, R] where G ≅ F/R and F is a free group. Hopf’s result led to the independent discovery of group cohomology by several groups in 1943-45: Eilenberg and Mac Lane in the USA (Rotman 1995, p. 358); Hopf and Eckmann in Switzerland; and Freudenthal in the Netherlands (Weibel 1999, p. 807). The situation was chaotic because communication between these countries was difficult during World War II. From a topological point of view, the homology and cohomology of G was first defined as the homology and cohomology of a model for the topological classifying space BG as discussed in #Connections with topological cohomology theories above. In practice, this meant using topology to produce the chain complexes used in formal algebraic definitions. From a module-theoretic point of view this was integrated into the Cartan–Eilenberg theory of homological algebra in the early 1950s. The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions). The cohomological part of class field theory was axiomatized as the theory of class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it) (Weibel 1999, p. 822). Some refinements in the theory post-1960 have been made, such as continuous cocycles and Tate's redefinition, but the basic outlines remain the same. This is a large field, and now basic in the theories of algebraic groups. The analogous theory for Lie algebras, called Lie algebra cohomology, was first developed in the late 1940s, by Chevalley–Eilenberg, and Koszul (Weibel 1999, p. 810). It is formally similar, using the corresponding definition of invariant for the action of a Lie algebra. It is much applied in representation theory, and is closely connected with the BRST quantization of theoretical physics. Group cohomology theory also has a direct application in condensed matter physics. Just like group theory being the mathematical foundation of spontaneous symmetry breaking phases, group cohomology theory is the mathematical foundation of a class of quantum states of matter—short-range entangled states with symmetry. Short-range entangled states with symmetry are also known as symmetry protected topological states.
100.8 Notes [1] Section VII.2 of Serre 1979 [2] Page 62 of Milne 2008 or section VII.3 of Serre 1979
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[3] Remark II.1.21 of Milne 2008
100.9 References • Adem, Alejandro; Milgram, R. James (2004), Cohomology of Finite Groups, Grundlehren der Mathematischen Wissenschaften 309 (2nd ed.), Springer-Verlag, ISBN 3-540-20283-8, MR 2035696, Zbl 1061.20044 • Brown, Kenneth S. (1972), Cohomology of Groups, Graduate Texts in Mathematics 87, Springer Verlag, ISBN 0-387-90688-6, MR 0672956 • Hopf, Heinz (1942), “Fundamentalgruppe und zweite Bettische Gruppe”, Comment. Math. Helv. 14 (1): 257–309, doi:10.1007/BF02565622, JFM 68.0503.01, MR 6510, Zbl 0027.09503 • Chapter II of Milne, James (5/2/2008), Class Field Theory, v4.00, retrieved 8/9/2008 Check date values in: |date=, |accessdate=, |year= / |date= mismatch (help) • Rotman, Joseph (1995), An Introduction to the Theory of Groups, Graduate Texts in Mathematics 148 (4th ed.), Springer-Verlag, ISBN 978-0-387-94285-8, MR 1307623 • Chapter VII of Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 554237, Zbl 0423.12016 • Serre, Jean-Pierre (1994), Cohomologie galoisienne, Lecture Notes in Mathematics 5 (Fifth ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-58002-7, MR 1324577 • Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, Princeton, NJ: Princeton University Press, ISBN 978-0-691-08017-8, MR 0347778 • Chapter 6 of Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR 1269324 • Weibel, Charles A. (1999), “History of homological algebra”, History of Topology, Cambridge University Press, pp. 797–836, ISBN 0-444-82375-1, MR 1721123
Chapter 101
Grunwald–Wang theorem In algebraic number theory, the Grunwald–Wang theorem is a local-global principle stating that—except in some precisely defined cases—an element x in a number field K is an nth power in K if it is an nth power in the completion Kp for all but finitely many primes p of K. For example, a rational number is a square of a rational number if it is a square of a p-adic number for almost all primes p. The Grunwald–Wang theorem is an example of a local-global principle. It was introduced by Wilhelm Grunwald (1933), but there was a mistake in this original version that was found and corrected by Shianghao Wang (1948). The theorem considered by Grunwald and Wang was more general than the one stated above as they discussed the existence of cyclic extensions with certain local properties, and the statement about nth powers is a consequence of this.
101.1 History Some days later I was with Artin in his office when Wang appeared. He said he had a counterexample to a lemma which had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself... Of course he [Artin] was astonished, as were all of us students, that a famous theorem with two published proofs, one of which we had all heard in the seminar without our noticing anything, could be wrong. John Tate, quoted in Roquette (2005, p.30) Grunwald (1933), a student of Hasse, gave an incorrect proof of the erroneous statement that an element in a number field is an nth power if it is an nth power locally almost everywhere. Whaples (1942) gave another incorrect proof of this incorrect statement. However Wang (1948) discovered the following counter-example: 16 is a p-adic 8th power for all odd primes p, but is not a rational or 2-adic 8th power. In his doctoral thesis Wang (1950) written under Artin, Wang gave and proved the correct formulation of Grunwald’s assertion, by describing the rare cases when it fails. This result is what is now known as the Grunwald–Wang theorem. The history of Wang’s counterexample is discussed in Roquette (2005, section 5.3)
101.2 Wang’s counter-example Grunwald’s original claim that an element that is an nth power almost everywhere locally is an nth power globally can fail in two distinct ways: the element can be an nth power almost everywhere locally but not everywhere locally, or it can be an nth power everywhere locally but not globally.
101.2.1
An element that is an nth power almost everywhere locally but not everywhere locally
The element 16 in the rationals is an 8th power at all places except 2, but is not an 8th power in the 2-adic numbers. 341
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It is clear that 16 is not a 2-adic 8th power, and hence not a rational 8th power, since the 2-adic valuation of 16 is 4 which is not divisible by 8. Generally, 16 is an 8th power in a field K if and only if the polynomial X 8 − 16 has a root in K. Write X 8 − 16 = (X 4 − 4)(X 4 + 4) = (X 2 − 2)(X 2 + 2)(X 2 − 2X + 2)(X 2 + 2X + 2). Thus, 16 is an 8th power in K if and only if 2, −2 or −1 is a square in K. Let p be any odd prime. It follows from the multiplicativity of the Legendre symbol that 2, −2 or −1 is a square modulo p. Hence, by Hensel’s lemma, 2, −2 or −1 is a square in Qp .
101.2.2
An element that is an nth power everywhere locally but not globally
√ √ 16 is not an 8th power in Q( 7) although √ it is an 8th √ power locally everywhere (i.e. in Qp ( 7) for all p). This follows from the above and the equality Q2 ( 7) = Q2 ( −1) .
101.3 A consequence of Wang’s counterexample Wang’s counterexample has the following interesting consequence showing that one cannot always find a cyclic Galois extension of a given degree of a number field in which finitely many given prime places split in a specified way: There exists no cyclic degree 8 extension K/Q in which the prime 2 is totally inert (i.e., such that K2 /Q2 is unramified of degree 8).
101.4 Special fields For any s ≥ 2 let ( ηs := exp
2πi 2s
)
( ) ( ) 2πi 2π + exp − s = 2 cos . 2 2s
Note that the 2s th cyclotomic field is
Q2s = Q(i, ηs ). A field is called s-special if it contains ηs , but neither i , ηs+1 nor iηs+1 .
101.5 Statement of the theorem Consider a number field K and a natural number n. Let S be a finite (possibly empty) set of primes of K and put
K(n, S) := {x ∈ K | x ∈ Kpn for all p ̸∈ S}. The Grunwald–Wang theorem says that
K(n, S) = K n unless we are in the special case which occurs when the following two conditions both hold: 1. K is s-special with an s such that 2s+1 divides n.
101.6. EXPLANATION OF WANG’S COUNTER-EXAMPLE
343
2. S contains the special set S0 consisting of those (necessarily 2-adic) primes p such that Kp is s-special. In the special case the failure of the Hasse principle is finite of order 2: the kernel of
K × /K ×n →
∏
Kp× /Kp×n
p̸∈S
is Z/2Z, generated by the element ηn s+1.
101.6 Explanation of Wang’s counter-example √ √ The field of rational numbers K = Q is 2-special since it contains η2 = 0 , but neither i , η3 = 2 nor iη3 = −2 . The special set is S0 = {2} . Thus, the special case in the Grunwald–Wang theorem occurs when n is divisible by 8, and S contains 2. This explains Wang’s counter-example and shows that it is minimal. It is also seen that an element in Q is an nth power if it is a p-adic nth power for all p. √ The field K = Q( 7) is 2-special as well, but with S0 = ∅ . This explains the other counter-example above.[1]
101.7 See also • The Hasse norm theorem states that for cyclic extensions an element is a norm if it is a norm everywhere locally.
101.8 Notes [1] See Chapter X of Artin–Tate.
101.9 References • Artin, Emil; Tate, John (1990), Class field theory, ISBN 978-0-8218-4426-7, MR 0223335 • Grunwald, W. (1933), “Ein allgemeiner Existenzsatz für algebraische Zahlkörper”, Journal für die reine und angewandte Mathematik 169: 103–107 • Roquette, Peter (2005), The Brauer-Hasse-Noether theorem in historical perspective, Schriften der MathematischNaturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften [Publications of the Mathematics and Natural Sciences Section of Heidelberg Academy of Sciences] 15, Berlin, New York: SpringerVerlag, ISBN 978-3-540-23005-2 • Wang, Shianghaw (1948), “A counter-example to Grunwald’s theorem”, Annals of Mathematics. Second Series 49: 1008–1009, ISSN 0003-486X, JSTOR 1969410, MR 0026992 • Wang, Shianghaw (1950), “On Grunwald’s theorem”, Annals of Mathematics. Second Series 51: 471–484, ISSN 0003-486X, JSTOR 1969335, MR 0033801 • Whaples, George (1942), “Non-analytic class field theory and Grünwald’s theorem”, Duke Mathematical Journal 9 (3): 455–473, doi:10.1215/s0012-7094-42-00935-9, ISSN 0012-7094, MR 0007010
Chapter 102
Hardy field In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that is closed under differentiation. They are named after the English mathematician G. H. Hardy.
102.1 Definition Initially at least, Hardy fields were defined in terms of germs of real functions at infinity. Specifically we consider a collection H of functions that are defined for all large real numbers, that is functions f that map (u,∞) to the real numbers R, where u is some real number depending on f. Here and in the rest of the article we say a function has a property “eventually” if it has the property for all sufficiently large x, so for example we say a function f in H is eventually zero if there is some real number U such that f(x) = 0 for all x ≥ U. We can form an equivalence relation on H by saying f is equivalent to g if and only if f − g is eventually zero. The equivalence classes of this relation are called germs at infinity. If H forms a field under the usual addition and multiplication of functions then so will H modulo this equivalence relation under the induced addition and multiplication operations. Moreover, if every function in H is eventually differentiable and the derivative of any function in H is also in H then H modulo the above equivalence relation is called a Hardy field.[1] Elements of a Hardy field are thus equivalence classes and should be denoted, say, [f]∞ to denote the class of functions that are eventually equal to the representative function f. However, in practice the elements are normally just denoted by the representatives themselves, so instead of [f]∞ one would just write f.
102.2 Examples If F is a subfield of R then we can consider it as a Hardy field by considering the elements of F as constant functions, that is by considering the number α in F as the constant function fα that maps every x in R to α. This is a field since F is, and since the derivative of every function in this field is 0 which must be in F it is a Hardy field. A less trivial example of a Hardy field is the field of rational functions on R, denoted R(x). This is the set of functions of the form P(x)/Q(x) where P and Q are polynomials with real coefficients. Since the polynomial Q can have only finitely many zeros by the fundamental theorem of algebra, such a rational function will be defined for all sufficiently large x, specifically for all x larger than the largest real root of Q. Adding and multiplying rational functions gives more rational functions, and the quotient rule shows that the derivative of rational function is again a rational function, so R(x) forms a Hardy field.
102.3 Properties Any element of a Hardy field is eventually either strictly positive, strictly negative, or zero. This follows fairly immediately from the facts that the elements in a Hardy field are eventually differentiable and hence continuous and 344
102.4. IN MODEL THEORY
345
eventually either have a multiplicative inverse or are zero. This means periodic functions such as the sine and cosine functions cannot exist in Hardy fields. This avoidance of periodic functions also means that every element in a Hardy field has a (possibly infinite) limit at infinity, so if f is an element of H, then
lim f (x)
x→∞
exists in R ∪ {−∞,+∞}.[2] It also means we can place an ordering on H by saying f < g if g − f is eventually strictly positive. Note that this is not the same as stating that f < g if the limit of f is less than the limit of g. For example if we consider the germs of the identity function f(x) = x and the exponential function g(x) = ex then since g(x) − f(x) > 0 for all x we have that g > f. But they both tend to infinity. In this sense the ordering tells us how quickly all the unbounded functions diverge to infinity.
102.4 In model theory The modern theory of Hardy fields doesn't restrict to real functions but to those defined in certain structures expanding real closed fields. Indeed, if R is an o-minimal expansion of a field, then the set of unary definable functions in R that are defined for all sufficiently large elements forms a Hardy field denoted H(R).[3] The properties of Hardy fields in the real setting still hold in this more general setting.
102.5 References [1] Boshernitzan, Michael (1986), “Hardy fields and existence of transexponential functions”, Aequationes Mathematicae 30 (1): 258–280, doi:10.1007/BF02189932 [2] Rosenlicht, Maxwell (1983), “The Rank of a Hardy Field”, Transactions of the American Mathematical Society 280 (2): 659–671, JSTOR 1999639 [3] Kuhlmann, Franz-Viktor; Kuhlmann, Salma (2003), “Valuation theory of exponential Hardy fields I”, Mathematische Zeitschrift 243 (4): 671–688, doi:10.1007/s00209-002-0460-4
Chapter 103
Hasse invariant of an algebra In mathematics, the Hasse invariant of an algebra is an invariant attached to a Brauer class of algebras over a field. The concept is named after Helmut Hasse. The invariant plays a role in local class field theory.
103.1 Local fields Let K be a local field with valuation v and D a K-algebra. We may assume D is a division algebra with centre K of degree n. The valuation v can be extended to D, for example by extending it compatibly to each commutative subfield of D: the value group of this valuation is (1/n)Z.[1] There is a commutative subfield L of D which is unramified over K, and D splits over L.[2] The field L is not unique but all such extensions are conjugate by the Skolem–Noether theorem, which further shows that any automorphism of L is induced by a conjugation in D. Take γ in D such that conjugation by γ induces the Frobenius automorphism of L/K and let v(γ) = k/n. Then k/n modulo 1 is the Hasse invariant of D. It depends only on the Brauer class of D.[3] The Hasse invariant is thus a map defined on the Brauer group of a local field K to the divisible group Q/Z.[3][4] Every class in the Brauer group is represented by a class in the Brauer group of an unramified extension of L/K of degree n,[5] which by the Grunwald–Wang theorem and the Albert–Brauer–Hasse–Noether theorem we may take to be a cyclic algebra (L,φ,πk ) for some k mod n, where φ is the Frobenius map and π is a uniformiser.[6] The invariant map attaches the element k/n mod 1 to the class. This exhibits the invariant map as a homomorphism
inv : Br(L/K) → Q/Z.
L/K
The invariant map extends to Br(K) by representing each class by some element of Br(L/K) as above.[3][4] For a non-Archimedean local field, the invariant map is a group isomorphism.[3][7] In the case of the field R of real numbers, there are two Brauer classes, represented by the algebra R itself and the quaternion algebra H.[8] It is convenient to assign invariant zero to the class of R and invariant 1/2 modulo 1 to the quaternion class. In the case of the field C of complex numbers, the only Brauer class is the trivial one, with invariant zero.[9]
103.2 Global fields For a global field K, given a central simple algebra D over K then for each valuation v of K we can consider the extension of scalars Dv = D ⊗ Kv The extension Dv splits for all but finitely many v, so that the local invariant of Dv is almost always zero. The Brauer group Br(K) fits into an exact sequence[8][9]
0 → Br(K) →
⊕
Br(Kv ) → Q/Z → 0,
v∈S
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103.3. REFERENCES
347
where S is the set of all valuations of K and the right arrow is the sum of the local invariants. The injectivity of the left arrow is the content of the Albert–Brauer–Hasse–Noether theorem. Exactness in the middle term is a deep fact from global class field theory.
103.3 References [1] Serre (1967) p.137 [2] Serre (1967) pp.130,138 [3] Serre (1967) p.138 [4] Lorenz (2008) p.232 [5] Lorenz (2008) pp.225–226 [6] Lorenz (2008) p.226 [7] Lorenz (2008) p.233 [8] Serre (1979) p.163 [9] Gille & Szamuely (2006) p.159
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001. • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 231–238. ISBN 978-0-387-72487-4. Zbl 1130.12001. • Serre, Jean-Pierre (1967). “VI. Local class field theory”. In Cassels, J.W.S.; Fröhlich, A. Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. London: Academic Press. pp. 128–161. Zbl 0153.07403. • Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics 67. Translated from the French by Marvin Jay Greenberg. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016.
103.4 Further reading • Shatz, Stephen S. (1972). Profinite groups, arithmetic, and geometry. Annals of Mathematics Studies 67. Princeton, NJ: Princeton University Press. ISBN 0-691-08017-8. MR 0347778. Zbl 0236.12002.
Chapter 104
Hasse norm theorem In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an element k of K such that there is an element l of L with NL/K (l) = k ; in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from LP; here the “prime” p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean. The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that √ √ 3 is a local norm everywhere for the extension Q( −3, √ √ 13)/Q but is not a global norm. Serre and Tate showed that another counterexample is given by the field Q( 13, 17)/Q where every rational square is a local norm everywhere but 52 is not a global norm. This is an example of a theorem stating a local-global principle. The full theorem is due to Hasse (1931). The special case when the degree n of the extension is 2 was proved by Hilbert (1897), and the special case when n is prime was proved by Furtwangler (1902). The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2 (L/K) is trivial if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the first cohomology of the idele class group vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2 (L/K) is isomorphic to the Tate cohomology group H0 (L/K) which describes which elements are norms, so for cyclic extensions it becomes Hasse’s theorem that an element is a norm if it is a local norm everywhere.
104.1 See also • Grunwald–Wang theorem, about when an element that is a power everywhere locally is a power.
104.2 References • Hasse, H. (1931), “Beweis eines Satzes und Wiederlegung einer Vermutung über das allgemeine Normenrestsymbol”, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 64–69 • H. Hasse, “A history of class field theory”, in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.XI. • G. Janusz, Algebraic number fields, Academic Press, 1973. Theorem V.4.5, p. 156
348
Chapter 105
Hasse principle In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
105.1 Intuition Given a polynomial equation with rational coefficients, if it has rational solution, then this also yields a real solution and a p-adic solution, as the rationals embed in the reals and p-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: when can you patch together solutions over the reals and p-adics to yield a solution over the rationals: when can local solutions be joined to form a global solution? One can ask this for other rings or fields: integers, for instance, or number fields. For number fields, rather than reals and p-adics, one uses complex embeddings and p -adics, for prime ideals p .
105.2 Forms representing 0 105.2.1
Quadratic forms
The Hasse–Minkowski theorem states that the local-global principle holds for the problem of representing 0 by quadratic forms over the rational numbers (which is Minkowski's result); and more generally over any number field (as proved by Hasse), when one uses all the appropriate local field necessary conditions. Hasse’s theorem on cyclic extensions states that the local-global principle applies to the condition of being a relative norm for a cyclic extension of number fields.
105.2.2
Cubic forms
A counterexample by Ernst S. Selmer shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3x3 + 4y3 + 5z3 = 0 has a solution in real numbers, and in all p-adic fields, but it has no nontrivial solution in which x, y, and z are all rational numbers.[1] Roger Heath-Brown showed[2] that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport.[3] Hence the local-global principle holds trivially for cubic forms over the rationals in at least 14 variables. If we confine ourselves to non-singular forms, one can do better than this: Heath-Brown proved that every nonsingular cubic form over the rational numbers in at least 10 variables represents 0,[4] thus trivially establishing the 349
350
CHAPTER 105. HASSE PRINCIPLE
Hasse principle for this class of forms. It is known that Heath-Brown’s result is best possible in the sense that there exist non-singular cubic forms over the rationals in 9 variables that don't represent zero.[5] However, Hooley showed that the Hasse principle holds for the representation of 0 by non-singular cubic forms over the rational numbers in at least nine variables.[6] Davenport, Heath-Brown and Hooley all used the Hardy–Littlewood circle method in their proofs. According to an idea of Manin, the obstructions to the Hasse principle holding for cubic forms can be tied into the theory of the Brauer group; this is the Brauer–Manin obstruction, which accounts completely for the failure of the Hasse principle for some classes of variety. However, Skorobogatov has shown that this is not the complete story.[7]
105.2.3
Forms of higher degree
Counterexamples by Fujiwara and Sudo show that the Hasse–Minkowski theorem is not extensible to forms of degree 10n + 5, where n is a non-negative integer.[8] On the other hand, Birch’s theorem shows that if d is any odd natural number, then there is a number N(d) such that any form of degree d in more than N(d) variables represents 0: the Hasse principle holds trivially.
105.3 Albert–Brauer–Hasse–Noether theorem The Albert–Brauer–Hasse–Noether theorem establishes a local-global principle for the splitting of a central simple algebra A over an algebraic number field K. It states that if A splits over every completion Kv then it is isomorphic to a matrix algebra over K.
105.4 Hasse principle for algebraic groups The Hasse principle for algebraic groups states that if G is a simply-connected algebraic group defined over the global field k then the map from
H 1 (k, G) →
∏
H 1 (ks , G)
s
is injective, where the product is over all places s of k. The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms. Kneser (1966) and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group E 8 which was only completed by Chernousov (1989) many years after the other cases. The Hasse principle for algebraic groups was used in the proofs of the Weil conjecture for Tamagawa numbers and the strong approximation theorem.
105.5 See also • Local analysis • Grunwald–Wang theorem • Grothendieck–Katz p-curvature conjecture
105.6 Notes [1] Ernst S. Selmer (1957). “The Diophantine equation ax3 + by3 + cz3 = 0”. Acta Mathematica 85: 203–362. doi:10.1007/BF02395746. [2] D.R. Heath-Brown (2007). “Cubic forms in 14 variables”. Invent. Math. 170: 199–230. doi:10.1007/s00222-007-0062-1.
105.7. REFERENCES
351
[3] H. Davenport (1963). “Cubic forms in sixteen variables”. Proceedings of the Royal Society A 272 (1350): 285–303. doi:10.1098/rspa.1963.0054. [4] D. R. Heath-Brown (1983). “Cubic forms in ten variables”. Proceedings of the London Mathematical Society 47 (2): 225–257. doi:10.1112/plms/s3-47.2.225. [5] L. J. Mordell (1937). “A remark on indeterminate equations in several variables”. Journal of the London Mathematical Society 12 (2): 127–129. doi:10.1112/jlms/s1-12.1.127. [6] C. Hooley (1988). “On nonary cubic forms”. J. Für die reine und angewandte Mathematik 386: 32–98. [7] Alexei N. Skorobogatov (1999). “Beyond the Manin obstruction”. Invent. Math. 135 (2): 399–424. doi:10.1007/s002220050291. [8] M. Fujiwara; M. Sudo (1976). “Some forms of odd degree for which the Hasse principle fails”. Pacific Journal of Mathematics 67 (1): 161–169. doi:10.2140/pjm.1976.67.161.
105.7 References • Chernousov, V. I. (1989), “The Hasse principle for groups of type E8”, Soviet Math. Dokl. 39: 592–596, MR 1014762 • Kneser, Martin (1966), “Hasse principle for H¹ of simply connected groups”, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 159–163, MR 0220736 • Serge Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 250–258. ISBN 3-540-61223-8. • Alexei Skorobogatov (2001). Torsors and rational points. Cambridge Tracts in Mathematics 144. Cambridge: Cambridge Univ. Press. pp. 1–7,112. ISBN 0-521-80237-7.
105.8 External links • Hazewinkel, Michiel, ed. (2001), “Hasse principle”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4 • PlanetMath article • Swinnerton-Dyer, Diophantine Equations: Progress and Problems, online notes
Chapter 106
Heegner number In number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q(√−d) has class number 1. Equivalently, its ring of integers has unique factorization.[1] The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the Stark–Heegner theorem there are precisely nine Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, 163 (OEIS sequence
A003173).
This result was conjectured by Gauss and proven by Kurt Heegner in 1952.
106.1 Euler’s prime-generating polynomial Euler’s prime-generating polynomial
n2 − n + 41, which gives (distinct) primes for n = 1, ..., 40, is related to the Heegner number 163 = 4 · 41 − 1. Euler’s formula, with n taking the values 1,... 40 is equivalent to
n2 + n + 41, with n taking the values 0,... 39, and Rabinowitz[2] proved that
n2 + n + p gives primes for n = 0, . . . , p − 2 if and only if its discriminant 1 − 4p equals minus a Heegner number. (Note that p − 1 yields p2 , so p − 2 is maximal.) 1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163 , yielding prime generating functions of Euler’s form for 2, 3, 5, 11, 17, 41 ; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.[3]
106.2 Almost integers and Ramanujan’s constant Ramanujan’s constant is the transcendental number[4] eπ to an integer:
√ 163
352
, which is an almost integer, in that it is very close
106.3. PI FORMULAS eπ
√
163
353
= 262 537 412 640 768 743.999 999 999 999 25... [5] ≈ 640 3203 + 744.
This number was discovered in 1859 by the mathematician Charles Hermite.[6] In a 1975 April Fool article in Scientific American magazine,[7] “Mathematical Games” columnist Martin Gardner made the (hoax) claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name. This coincidence is explained by complex multiplication and the q-expansion of the j-invariant.
106.2.1
Detail
Briefly, j((1 + q-expansion.
√ √ √ −d)/2) is an integer for d a Heegner number, and eπ d ≈ −j((1 + −d)/2) + 744 via the
If τ is a quadratic irrational, then the j-invariant is an algebraic integer of degree |Cl(Q(τ ))| , the class number of Q(τ ) and the minimal (monic integral) polynomial it satisfies is called the Hilbert class polynomial. Thus if the imaginary quadratic extension Q(τ ) has class number 1 (so d is a Heegner number), the j-invariant is an integer. The q-expansion of j, with its Fourier series expansion written as a Laurent series in terms of q = exp(2πiτ ) , begins as:
j(q) =
1 + 744 + 196 884q + · · · . q
√ The coefficients cn asymptotically grow as ln(cn ) ∼ 4π n + O(ln(n)) , and the low order coefficients grow more slowly√than 200 000n , so for q ≪ 1/200 √ 000 , j is very well approximated√by its first two terms.√ Setting τ = (1 + −163)/2 yields q = − exp(−π 163) or equivalently, 1q = − exp(π 163) . Now j((1 + −163)/2) = (−640 320)3 , so,
(−640 320)3 = −eπ
√ 163
( ) √ + 744 + O e−π 163 .
Or,
eπ
√ 163
( ) √ = 640 3203 + 744 + O e−π 163
where the linear term of the error is,
−196 884/eπ
√ 163
explaining why eπ
≈ 196 884/(640 3203 + 744) ≈ −0.000 000 000 000 75
√ 163
is within approximately the above of being an integer.
106.3 Pi formulas The Chudnovsky brothers found in 1987,
∞ ∑ 12 (6k)!(163 · 3 344 418k + 13 591 409) 1 = 3/2 π (3k)!(k!)3 (−640 320)3k 640 320 k=0
and uses the fact that j
( 1+√−163 ) 2
= −640 3203 . For similar formulas, see the Ramanujan–Sato series.
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CHAPTER 106. HEEGNER NUMBER
106.4 Other Heegner numbers For the four largest Heegner numbers, the approximations one obtains[8] are as follows.
eπ e e e
√ 19
√ π 43
√ π 67
√ π 163
≈ 963 + 744 − 0.22 ≈ 9603 + 744 − 0.000 22 ≈ 5 2803 + 744 − 0.000 0013 ≈ 640 3203 + 744 − 0.000 000 000 000 75
Alternatively,[9]
eπ e e e
√ 19
√ π 43 √ π 67
√ π 163
≈ 123 (32 − 1)3 + 744 − 0.22 ≈ 123 (92 − 1)3 + 744 − 0.000 22 ≈ 123 (212 − 1)3 + 744 − 0.000 0013 ≈ 123 (2312 − 1)3 + 744 − 0.000 000 000 000 75
where the reason for the squares is due to certain Eisenstein series. For Heegner numbers d < 19 , one does not obtain an almost integer; even d = 19 is not noteworthy.[10] The integer j-invariants are highly factorisable, which follows from the 123 (n2 − 1)3 = (22 · 3 · (n − 1) · (n + 1))3 form, and factor as, √ −19)/2) = 963 = (25 · 3)3 √ j((1 + −43)/2) = 9603 = (26 · 3 · 5)3 √ j((1 + −67)/2) = 5 2803 = (25 · 3 · 5 · 11)3 √ j((1 + −163)/2) = 640 3203 = (26 · 3 · 5 · 23 · 29)3 . j((1 +
These transcendental numbers, in addition to being closely approximated by integers, (which are simply algebraic numbers of degree 1), can also be closely approximated by algebraic numbers of degree 3,[11]
eπ e e e
√ 19
√ π 43 √ π 67
√ π 163
≈ x24 − 24; x3 − 2x − 2 = 0 ≈ x24 − 24; x3 − 2x2 − 2 = 0 ≈ x24 − 24; x3 − 2x2 − 2x − 2 = 0 ≈ x24 − 24; x3 − 6x2 + 4x − 2 = 0
The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation. In addition, they can also be closely approximated by algebraic numbers of degree 4,[12]
eπ e
√ 19
√ π 43
eπ eπ
√ 67
√ 163
( )−2 √ √ ≈ 35 3 − 2(−3 + 1 3 · 19) − 12.000 06 . . . ( )−2 √ √ ≈ 3 9 − 2(−39 + 7 3 · 43) − 12.000 000 061 . . . 5
( )−2 √ √ ≈ 35 21 − 2(−219 + 31 3 · 67) − 12.000 000 000 36 . . . ( )−2 √ √ ≈ 35 231 − 2(−26 679 + 2 413 3 · 163) − 12.000 000 000 000 000 21 . . .
Note the reappearance of the integers n = 3, 9, 21, 231 as well as the fact that,
106.5. CONSECUTIVE PRIMES
355
26 · 3(−32 + 3 · 19 · 12 ) = 962 26 · 3(−392 + 3 · 43 · 72 ) = 9602 26 · 3(−2192 + 3 · 67 · 312 ) = 5 2802 26 · 3(−266792 + 3 · 163 · 24132 ) = 640 3202 which, with the appropriate fractional power, are precisely the j-invariants. As well as for algebraic numbers of degree 6,
eπ eπ e e
√ 19 √ 43
√ π 67
√ π 163
≈ (5x)3 − 6.000 010 . . . ≈ (5x)3 − 6.000 000 010 . . . ≈ (5x)3 − 6.000 000 000 061 . . . ≈ (5x)3 − 6.000 000 000 000 000 034 . . .
where the xs are given respectively by the appropriate root of the sextic equations, 5x6 − 96x5 − 10x3 + 1 = 0 5x6 − 960x5 − 10x3 + 1 = 0 5x6 − 5 280x5 − 10x3 + 1 = 0 5x6 − 640 320x5 − 10x3 + 1 = 0 with the j-invariants appearing again. These √sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension Q 5 (with the first factoring further into two quadratics). These √ algebraic approximations can be exactly expressed in terms of Dedekind eta quotients. As an example, let τ = (1+ −163)/2 , then,
eπ eπ eπ
√ 163 √ 163 √ 163
( = (
eπi/24 η(τ ) η(2τ )
)24 − 24.000 000 000 000 001 05 . . .
)12 eπi/12 η(τ ) − 12.000 000 000 000 000 21 . . . η(3τ ) ( πi/6 )6 e η(τ ) = − 6.000 000 000 000 000 034 . . . η(5τ ) =
where the eta quotients are the algebraic numbers given above.
106.5 Consecutive primes Given an odd prime p, if one computes k 2 (mod p) for k = 0, 1, . . . , (p − 1)/2 (this is sufficient because (p − k)2 ≡ k 2 (mod p) ), one gets consecutive composites, followed by consecutive primes, if and only if p is a Heegner number.[13] For details, see “Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields” by Richard Mollin.
106.6 Notes and references [1] Conway, John Horton; Guy, Richard K. (1996). The Book of Numbers. Springer. p. 224. ISBN 0-387-97993-X. [2] Rabinowitz, G. “Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern.” Proc. Fifth Internat. Congress Math. (Cambridge) 1, 418–421, 1913.
356
CHAPTER 106. HEEGNER NUMBER
[3] Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983. √
[4] Weisstein, Eric W., “Transcendental Number”, MathWorld. gives eπ d , d ∈ Z ∗ , based on Nesterenko, Yu. V. “On Algebraic Independence of the Components of Solutions of a System of Linear Differential Equations.” Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495–512, 1974. English translation in Math. USSR 8, 501–518, 1974. [5] Ramanujan Constant – from Wolfram MathWorld [6] Barrow, John D (2002). The Constants of Nature. London: Jonathan Cape. ISBN 0-224-06135-6. [7] Gardner, Martin (April 1975). “Mathematical Games”. Scientific American (Scientific American, Inc) 232 (4): 127. √ √ √ 3 [8] These can be checked by computing eπ d − 744 on a calculator, and 196 884/eπ d for the linear term of the error. [9] http://groups.google.com.ph/group/sci.math.research/browse_thread/thread/3d24137c9a860893?hl=en# [10] The absolute deviation of a random real number (picked uniformly from [0,1], say) is a uniformly distributed variable on [0, 0.5], so it has absolute average deviation and median absolute deviation of 0.25, and a deviation of 0.22 is not exceptional. [11] “Pi Formulas”. [12] “Extending Ramanujan’s Dedekind Eta Quotients”. [13] http://www.mathpages.com/home/kmath263.htm
106.7 External links • Weisstein, Eric W., “Heegner Number”, MathWorld. • "Sloane’s A003173 : Heegner numbers: imaginary quadratic fields with unique factorization", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. • Gauss’ Class Number Problem for Imaginary Quadratic Fields, by Dorian Goldfeld: Detailed history of problem. • Clark, Alex. “163 and Ramanujan Constant”. Numberphile. Brady Haran.
Chapter 107
Heegner point In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss’s conjecture on imaginary quadratic fields of class number one. The Gross–Zagier theorem (Gross & Zagier 1986) describes the height of Heegner points in terms of a derivative of the L-function of the elliptic curve at the point s = 1. In particular if the elliptic curve has (analytic) rank 1, then the Heegner points can be used to construct a rational point on the curve of infinite order (so the Mordell–Weil group has rank at least 1). More generally, Gross, Kohnen & Zagier (1987) showed that Heegner points could be used to construct rational points on the curve for each positive integer n, and the heights of these points were the coefficients of a modular form of weight 3/2. Kolyvagin later used Heegner points to construct Euler systems, and used this to prove much of the Birch–SwinnertonDyer conjecture for rank 1 elliptic curves. Shouwu Zhang generalized the Gross–Zagier theorem from elliptic curves to the case of modular abelian varieties. Brown proved the Birch–Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic. (Brown 1994) Heegner points can be used to compute very large rational points on rank 1 elliptic curves (see (Watkins 2006) for a survey) that could not be found by naive methods. Implementation of the algorithm is available in Magma and PARI/GP
107.1 References • Birch, B., “Heegner points: the beginnings”, in Darmon, Henri; Zhang, Shou-wu, Heegner Points and Rankin L-Series, Mathematical Sciences Research Institute Publications 49, Cambridge University Press, ISBN 0-52183659-X, MR 2083207. • Brown, M. L. (2004), Heegner modules and elliptic curves, Lecture Notes In Mathematics 1849, SpringerVerlag, ISBN 3-540-22290-1, MR 2082815. • Darmon, Henri; Zhang, Shou-Wu, eds. (2004), Heegner points and Rankin L-series, Mathematical Sciences Research Institute Publications 49, Cambridge University Press, ISBN 978-0-521-83659-3, MR 2083206 • Gross, Benedict H.; Zagier, Don B. (1986), “Heegner points and derivatives of L-series”, Inventiones Mathematicae 84 (2): 225–320, doi:10.1007/BF01388809, MR 0833192. • Gross, B.; Kohnen, W.; Zagier, D. (1987), “Heegner points and derivatives of L-series. II”, Mathematische Annalen 278 (1–4): 497–562, doi:10.1007/BF01458081, MR 0909238. • Heegner, Kurt (1952), “Diophantische Analysis und Modulfunktionen”, Mathematische Zeitschrift 56 (3): 227– 253, doi:10.1007/BF01174749, MR 0053135. • Watkins, Mark (2006), Some remarks on Heegner point computations, arXiv:math/0506325v2. • Brown, Mark (1994), “On a conjecture of Tate for elliptic surfaces over finite fields”, Proc. London Math. Soc. 69 (3): 489–514.
357
Chapter 108
Herbrand quotient In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.
108.1 Definition If G is a finite cyclic group acting on a G-module A, then the cohomology groups H n (G,A) have period 2 for n≥1; in other words H n (G,A) = H n+2 (G,A), an isomorphism induced by cup product with a generator of H 2 (G,Z). (If instead we use the Tate cohomology groups then the periodicity extends down to n=0.) A Herbrand module is an A for which the cohomology groups are finite. In this case, the Herbrand quotient h(G,A) is defined to be the quotient h(G,A) = |H 2 (G,A)|/|H 1 (G,A)| of the order of the even and odd cohomology groups.
108.1.1
Alternative definition
The quotient may be defined for a pair of endomorphisms of an Abelian group, f and g, which satisfy the condition fg = gf = 0. Their Herbrand quotient q(f,g) is defined as
q(f, g) =
|kerf : img| |kerg : imf |
if the two indices are finite. If G is a cyclic group with generator γ acting on an Abelian group A, then we recover the previous definition by taking f = 1 - γ and g = 1 + γ + γ2 + ... .
108.2 Properties • The Herbrand quotient is multiplicative on short exact sequences.[1] In other words, if 0→A→B→C→0 is exact, and any two of the quotients are defined, then so is the third and[2] 358
108.3. SEE ALSO
359
h(G,B) = h(G,A)h(G,C) • If A is finite then h(G,A) = 1.[2] • For A is a submodule of the G-module B of finite index, if either quotient is defined then so is the other and they are equal:[1] more generally, if there is a G-morphism A → B with finite kernel and cokernel then the same holds.[2] • If Z is the integers with G acting trivially, then h(G,Z) = |G| • If A is a finitely generated G-module, then the Herbrand quotient h(A) depends only on the complex G-module C⊗A (and so can be read off from the character of this complex representation of G). These properties mean that the Herbrand quotient is usually relatively easy to calculate, and is often much easier to calculate than the orders of either of the individual cohomology groups.
108.3 See also • Class formation
108.4 References [1] Cohen (2007) p.245 [2] Serre (1979) p.134
• Atiyah, M.F.; Wall, C.T.C. (1967). “Cohomology of Groups”. In Cassels, J.W.S.; Fröhlich, Albrecht. Algebraic Number Theory. Academic Press. Zbl 0153.07403. See section 8. • Artin, Emil; Tate, John (2009). Class Field Theory. AMS Chelsea. p. 5. ISBN 0-8218-4426-1. Zbl 1179.11040. • Cohen, Henri (2007). Number Theory – Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics 239. Springer-Verlag. pp. 242–248. ISBN 978-0-387-49922-2. Zbl 1119.11001. • Janusz, Gerald J. (1973). Algebraic number fields. Pure and Applied Mathematics 55. Academic Press. p. 142. Zbl 0307.12001. • Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). SpringerVerlag. pp. 120–121. ISBN 3-540-63003-1. Zbl 0819.11044. • Serre, Jean-Pierre (1979). Local fields. Graduate Texts in Mathematics 67. Translated from the French by Marvin Jay Greenberg. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016.
Chapter 109
Hermite’s problem Hermite’s problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational.
109.1 Motivation A standard way of writing real numbers is by their decimal representation, such as:
x = a0 .a1 a2 a3 . . . where a0 is an integer, the integer part of x, and a1 , a2 , a3 … are integers between 0 and 9. Given this representation the number x is equal to
x=
∞ ∑ an . n 10 n=0
The real number x is a rational number only if its decimal expansion is eventually periodic, that is if there are natural numbers N and p such that for every n ≥ N it is the case that an₊p = an. Another way of expressing numbers is to write them as continued fractions, as in:
x = [a0 ; a1 , a2 , a3 , . . .], where a0 is an integer and a1 , a2 , a3 … are natural numbers. From this representation we can recover x since
1
x = a0 +
.
1
a1 + a2 +
1 . a3 + . .
If x is a rational number then the sequence (an) terminates after finitely many terms. On the other hand, Euler proved that irrational numbers require an infinite sequence to express them as continued fractions.[1] Moreover, this sequence is eventually periodic (again, so that there are natural numbers N and p such that for every n ≥ N we have an₊p = an), if and only if x is a quadratic irrational. 360
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361
109.2 Hermite’s question Rational numbers are algebraic numbers that satisfy a polynomial of degree 1, while quadratic irrationals are algebraic numbers that satisfy a polynomial of degree 2. For both these sets of numbers we have a way to construct a sequence of natural numbers (an) with the property that each sequence gives a unique real number and such that this real number belongs to the corresponding set if and only if the sequence is eventually periodic. In 1848 Charles Hermite wrote a letter to Carl Gustav Jacob Jacobi asking if this situation could be generalised, that is can one assign a sequence of natural numbers to each real number x such that the sequence is eventually periodic precisely when x is a cubic irrational, that is an algebraic number of degree 3?[2][3] Or, more generally, for each natural number d is there a way of assigning a sequence of natural numbers to each real number x that can pick out when x is algebraic of degree d?
109.3 Approaches Sequences that attempt to solve Hermite’s problem are often called multidimensional continued fractions. Jacobi himself came up with an early example, finding a sequence corresponding to each pair of real numbers (x,y) that acted as a higher-dimensional analogue of continued fractions.[4] He hoped to show that the sequence attached to (x, y) was eventually periodic if and only if both x and y belonged to a cubic number field, but was unable to do so and whether this is the case remains unsolved. In 2015, for the first time, a periodic representation for any cubic irrational has been provided by means of ternary continued fractions, i.e., the problem of writing cubic irrationals as a periodic sequence of rational or integer numbers has been solved. However, the periodic representation does not derive from an algorithm defined over all real numbers and it is derived only starting from the knowledge of the minimal polynomial of the cubic irrational.[5] Rather than generalising continued fractions, another approach to the problem is to generalise Minkowski’s question mark function. This function ? : [0, 1] → [0, 1] also picks out quadratic irrational numbers since ?(x) is rational if and only if x is either rational or a quadratic irrational number, and moreover x is rational if and only if ?(x) is a dyadic rational, thus x is a quadratic irrational precisely when ?(x) is a non-dyadic rational number. Various generalisations of this function to either the unit square [0, 1] × [0, 1] or the two-dimensional simplex have been made, though none has yet solved Hermite’s problem.[6][7]
109.4 References [1] “E101 – Introductio in analysin infinitorum, volume 1”. Retrieved 2008-03-16. [2] Émile Picard, L'œuvre scientifique de Charles Hermite, Ann. Sci. École Norm. Sup. 3 18 (1901), pp.9–34. [3] Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objects de la théorie des nombres. (Continuation)., Journal für die reine und angewandte Mathematik 40 (1850), pp.279–315, doi:10.1515/crll.1850.40.279 [4] C. G. J. Jacobi, Allgemeine Theorie der kettenbruchänlichen Algorithmen, in welche jede Zahl aus drei vorhergehenden gebildet wird (English: General theory of continued-fraction-like algorithms in which each number is formed from three previous ones), Journal für die reine und angewandte Mathematik 69 (1868), pp.29–64. [5] Nadir Murru, On the periodic writing of cubic irrationals and a generalization of Rédei functions, Int. J. Number Theory 11 (2015), no. 3, pp. 779-799, doi: 10.1142/S1793042115500438 [6] L. Kollros, Un Algorithme pour l'approximation simultanée de Deux Granduers, Inaugural-Dissertation, Universität Zürich, 1905. [7] Olga R. Beaver, Thomas Garrity, A two-dimensional Minkowski ?(x) function, J. Number Theory 107 (2004), no. 1, pp. 105–134.
Chapter 110
Higher local field In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields. The concept was introduced by A. N. Parshin and K. Kato in the 1970s.[1] On the usual local fields (typically completions of number fields or the quotient fields of local rings of algebraic curves) there is a unique surjective discrete valuation (of rank 1) associated to a choice of a local parameter of the fields, unless they are archimedean local fields such as the real numbers and complex numbers. Similarly, there is a discrete valuation of rank n on almost all n-dimensional local fields, associated to a choice of n local parameters of the field.[2] In contrast to one-dimensional local fields, higher local fields have a sequence of residue fields.[3] There are different integral structures on higher local fields, depending how many residue fields information one wants to take into account.[3] Geometrically, higher local fields appear via a process of localization and completion of local rings of higher dimensional schemes.[3] Higher local fields are an important part of the subject of higher dimensional number theory, forming the appropriate collection of objects for local considerations.
110.1 Definition Any complete discrete valuation field has a (possibly infinite) complete discrete valuation dimension, which can be defined inductively as follows. Finite fields have dimension 0 and complete discrete valuation fields with finite residue field have dimension one (it is natural to also define archimedean local fields such as R or C to have dimension 1), then we say a complete discrete valuation field has dimension n if its residue field has dimension n−1. Higher local fields are those of dimension greater than 1, it is clear that one-dimensional local fields are the traditional local fields. We call the residue field of a finite-dimensional higher local field the 'first' residue field, its residue field is then the second residue field, and the pattern continues until we reach a finite field.[3]
110.2 Examples Two-dimensional local fields are divided into the following classes: • Fields of positive characteristic, they are formal power series in variable u over a one-dimensional local field, i.e. Fq((u))((v)). • Equi-characteristic fields of characteristic zero, they are formal power series F((u)) over a one-dimensional local field F of characteristic zero. • Mixed-characteristic fields, they are finite extensions of fields of type F{{t}}, F is a one-dimensional local field of characteristic zero. This field is defined as the set of formal power series, infinite in both directions, with coefficients from F such that the minimum of the valuation of the coefficients is an integer, and such that the valuation of the coefficients tend to zero as their index goes to minus infinity.[3] 362
110.3. CONSTRUCTIONS
363
• Archimedean two-dimensional local fields, which are formal power series over the real numbers R or the complex numbers C.
110.3 Constructions Higher local fields appear in a variety of contexts. A geometric example is as follows. Given a surface over a finite field of characteristic p, a curve on the surface and a point on the curve, take the local ring at the point. Then, complete this ring, localise it at the curve and complete the resulting ring. Finally, take the quotient field. The result is a two-dimensional local field over a finite field.[3] There is also a construction using commutative algebra, which becomes technical for non-regular rings. The starting point is a Noetherian, regular, n-dimensional ring and a full flag of prime ideals such that their corresponding quotient ring is regular. A series of completions and localisations take place as above until an n-dimensional local field is reached.[4]
110.4 Topology One-dimensional local fields are usually considered in the valuation topology, in which the discrete valuation is used to define open sets. This will not suffice for higher dimensional local fields, since one needs to take into account the topology at the residue level too. Higher local fields can be endowed with appropriate topologies which address this issue. Such topologies are not the topologies associated with discrete valuations of rank n, if n > 1. In dimension two and higher the additive group of the field becomes a topological group which is not locally compact and the base of the topology is not countable. The most surprising thing is that the multiplication is not continuous, however, it is sequentially continuous which suffices for all reasonable arithmetic purposes. There are also iterated ind pro approaches to replace topological considerations by more formal ones.[5]
110.5 Measure, integration and harmonic analysis on higher local fields There is no translation invariant measure on two-dimensional local fields. Instead, there is a finitely additive translation invariant measure defined on the ring of sets generated by closed balls with respect to two-dimensional discrete valuations on the field, and taking values in formal power series R((X)) over reals.[6] This measure is also countably additive in a certain refined sense. It can be viewed as higher Haar measure on higher local fields. The additive group of every higher local field is non-canonically self-dual, and one can define a higher Fourier transform on appropriate spaces of functions. This leads to higher harmonic analysis. The Fourier transform has a number of features similar to those of the Feynman path integral.[7]
110.6 Class field theory Local class field theory in dimension one has its analogues in higher dimensions. The appropriate replacement for the multiplicative group becomes the nth Milnor K-group, where n is the dimension of the field, which then appears as the domain of a reciprocity map to the Galois group of the maximal abelian extension over the field. Even better is to work with the quotient of the nth Milnor K-group by its subgroup of elements divisible by any positive integer, this quotient can also be viewed as the maximal separated topological quotient of the K-group endowed with appropriate higher dimensional topology. Higher local class field theory is compatible with class field theory at the residue field level, using the border map of Milnor K-theory to create a commutative diagram involving the reciprocity map on the level of the field and the residue field.[8] Higher local class field theory in positive characteristic was proposed by A.N. Parshin, full higher local class field theory was developed by Kazuya Kato, and an alternative full explicit higher local class field theory was constructed by Ivan Fesenko.[5]
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110.7 Higher adeles Unlike the classical case of dimension one, there are two different adelic structures on arithmetic schemes in dimension two: one of them is of more geometric origin and is well suited for such issues as the study of 1-cycles, intersection theory, line and vector bundles, etc. Another is of more analytic origin and is well suited for integration and such issues as zeta integrals and the study of the zeta function of the scheme. Interaction between the multiplicative groups of the two adelic structures is part of higher class field theory. Higher local fields participate as components of both higher adeles, the definition of which involves a product over full flags for the scheme.[9]
110.8 Notes [1] Osipov (2008) p.131 [2] Fesenko, I.B., Vostokov, S.V. Local Fields and Their Extensions. American Mathematical Society, 1992, Chapter 1 and Appendix. [3] Fesenko, I., Kurihara, M. (eds.) Invitation to Higher Local Fields. Geometry and Topology Monographs, 2000, section 1 (Zhukov). [4] Morrow, M., Introduction to Higher Local Fields. Preprint, University of Chicago 2011 [5] Fesenko, I., Kurihara, M. (eds.) Invitation to Higher Local Fields. Geometry and Topology Monographs, 2000, several sections. [6] Fesenko, I. Analysis on arithmetic schemes. I. Docum. Math., (2003), Kato’s special volume, 261-284 [7] Fesenko, I., Measure, integration and elements of harmonic analysis on generalized loop spaces, Proceed. St. Petersburg Math. Soc., vol. 12 (2005), 179-199; AMS Transl. Series 2, vol. 219, 149-164, 2006 [8] Fesenko, I., Kurihara, M. (eds.) Invitation to Higher Local Fields. Geometry and Topology Monographs, 2000, section 5 (Kurihara). [9] Osipov (2008) pp.131-164
110.9 References • Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3259-2, MR 1915966, Zbl 1156.11046 • Fesenko, Ivan B.; Kurihara, Masato, eds. (2000), Invitation to Higher Local Fields. Extended version of talks given at the conference on higher local fields, Münster, Germany, August 29–September 5, 1999, Geometry and Topology Monographs 3 (First ed.), University of Warwick: Mathematical Sciences Publishers, doi:10.2140/gtm.2000.3, ISSN 1464-8989, Zbl 0954.00026 • Osipov, Denis V. (2008), "n-dimensional local fields and adeles on n-dimensional schemes”, in Young, Nicholas; Choi, Yemon, Surveys in contemporary mathematics, London Mathematical Society Lecture Note Series 347, Cambridge University Press, pp. 131–164, ISBN 978-0-521-70564-6, Zbl 1144.11078
110.10 External links • Local Fields and Their Extensions, Second extended edition • Invitation to Higher Local Fields • Introduction to Higher Local Fields
Chapter 111
Hilbert class field In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K. In this context, the Hilbert class field of K is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E (rather than to a complex embedding of E).
111.1 Examples • If the ring of integers of K is a unique factorization domain, in particular, if K = Q then K is its own Hilbert class field. √ √ √ • Let K = Q( −15) of discriminant 15. The field L = Q( −3, 5) has discriminant 225=152 and so is an everywhere unramified extension of K, and it is abelian. Using the Minkowski bound, one can show that K has class number 2. Hence, its Hilbert class field is L . A non-principal ideal of K is (2,(1+√−15)/2), and in L this becomes the principal ideal ((1+√5)/2). • To see why ramification at the archimedean primes must be taken into account, consider the real quadratic field K obtained by adjoining the square root of 3 to Q. This field has class number 1 and discriminant 3, but the extension K(i)/K of discriminant 9=32 is unramified at all prime ideals in K, so K admits finite abelian extensions of degree greater than 1 in which all finite primes of K are unramified. This doesn't contradict the Hilbert class field of K being K itself: every proper finite abelian extension of K must ramify at some place, and in the extension K(i)/K there is ramification at the archimedean places: the real embeddings of K extend to complex (rather than real) embeddings of K(i). • By the theory of complex multiplication, the Hilbert class field of an imaginary quadratic field is generated by the value of the elliptic modular function at a generator for the ring of integers (as a Z-module).
111.2 History The existence of a (narrow) Hilbert class field for a given number field K was conjectured by David Hilbert (1902) and proved by Philipp Furtwängler.[1] The existence of the Hilbert class field is a valuable tool in studying the structure of the ideal class group of a given field.
111.3 Additional properties The Hilbert class field E also satisfies the following: 365
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• E is a finite Galois extension of K and [E : K]=hK, where hK is the class number of K. • The ideal class group of K is isomorphic to the Galois group of E over K. • Every ideal of OK is a principal ideal of the ring extension OE (principal ideal theorem). • Every prime ideal P of OK decomposes into the product of hK/f prime ideals in OE, where f is the order of [P] in the ideal class group of OK. In fact, E is the unique field satisfying the first, second, and fourth properties.
111.4 Explicit constructions If K is imaginary quadratic and A is an elliptic curve with complex multiplication by the ring of integers of K, then adjoining the j-invariant of A to K gives the Hilbert class field.[2]
111.5 Generalizations In class field theory, one studies the ray class field with respect to a given modulus, which is a formal product of prime ideals (including, possibly, archimedean ones). The ray class field is the maximal abelian extension unramified outside the primes dividing the modulus and satisfying a particular ramification condition at the primes dividing the modulus. The Hilbert class field is then the ray class field with respect to the trivial modulus 1. The narrow class field is the ray class of all infinite primes. For example, √ field with respect to the modulus consisting √ the argument above shows that Q( 3, i) is the narrow class field of Q( 3) .
111.6 Notes [1] Furtwängler 1906 [2] Theorem II.4.1 of Silverman 1994
111.7 References • Childress, Nancy (2009), Class field theory, New York: Springer, doi:10.1007/978-0-387-72490-4, ISBN 978-0-387-72489-8, MR 2462595 • Furtwängler, Philipp (1906), “Allgemeiner Existenzbeweis für den Klassenkörper eines beliebigen algebraischen Zahlkörpers”, Mathematische Annalen 63 (1): 1–37, doi:10.1007/BF01448421, JFM 37.0243.02, MR 1511392, retrieved 2009-08-21 • Hilbert, David (1902) [1898], "Über die Theorie der relativ-Abel’schen Zahlkörper”, Acta Mathematica 26 (1): 99–131, doi:10.1007/BF02415486 • J. S. Milne, Class Field Theory (Course notes available at http://www.jmilne.org/math/). See the Introduction chapter of the notes, especially p. 4. • Silverman, Joseph H. (1994), Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151, New York: Springer-Verlag, ISBN 978-0-387-94325-1 • Gras, Georges (2005), Class field theory: From theory to practice, New York: Springer This article incorporates material from Existence of Hilbert class field on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Chapter 112
Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K × × K × to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers . It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by David Hilbert (1897, sections 64, 131, 1998, English translation) in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields. The Hilbert symbol has been generalized to higher local fields.
112.1 Quadratic Hilbert symbol Over a local field K whose multiplicative group of non-zero elements is K × , the quadratic Hilbert symbol is the function (–, –) from K × × K × to {−1,1} defined by { (a, b) =
112.1.1
1, −1,
if z 2 = ax2 + by 2 has a non-zero solution (x, y, z) ∈ K 3 ; if not.
Properties
The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above: • If a is a square, then (a, b) = 1 for all b. • For all a,b in K × , (a, b) = (b, a). • For any a in K × such that a−1 is also in K × , we have (a, 1−a) = 1. The (bi)multiplicativity, i.e., (a, b1 b2 ) = (a, b1 )·(a, b2 ) for any a, b1 and b2 in K × is, however, more difficult to prove, and requires the development of local class field theory. The third property shows that the Hilbert symbol is an example of a Steinberg symbol and thus factors over the second Milnor K-group K2M (K) , which is by definition K × ⊗ K × / (a ⊗ (1−a), a ∈ K × \ {1}) By the first property it even factors over K2M (K)/2 . This is the first step towards the Milnor conjecture. 367
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112.1.2
Interpretation as an algebra
The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules i2 = a , j 2 = b , ij = −ji = k . In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with −1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
112.1.3
Hilbert symbols over the rationals
For a place v of the rational number field and rational numbers a, b we let (a, b)v denote the value of the Hilbert symbol in the corresponding completion Qv. As usual, if v is the valuation attached to a prime number p then the corresponding completion is the p-adic field and if v is the infinite place then the completion is the real number field. Over the reals, (a, b)∞ is +1 if at least one of a or b is positive, and −1 if both are negative. Over the p-adics with p odd, writing a = pα u and b = pβ v , where u and v are integers coprime to p, we have (a, b)p = (−1)αβϵ(p)
( )β ( )α u p
v p
, where ϵ(p) = (p − 1)/2
and the expression involves two Legendre symbols. Over the 2-adics, again writing a = 2α u and b = 2β v , where u and v are odd numbers, we have (a, b)2 = (−1)ϵ(u)ϵ(v)+αω(v)+βω(u) , where ω(x) = (x2 − 1)/8. It is known that if v ranges over all places, (a, b)v is 1 for almost all places. Therefore the following product formula ∏ (a, b)v = 1 v
makes sense. It is equivalent to the law of quadratic reciprocity.
112.1.4
Kaplansky radical
The Hilbert symbol on a field F defines a map
(·, ·) : F ∗ /F ∗2 × F ∗ /F ∗2 → Br(F ) where Br(F) is the Brauer group of F. The kernel of this mapping, the elements a such that (a,b)=1 for all b, is the Kaplansky radical of F.[1] The radical is a subgroup of F* /F*2 , identified with a subgroup of F* . The radical contains is equal to F* if and only if F is not formally real and has u-invariant at most 2.[2] In the opposite direction, a field with radical F*2 is termed a Hilbert field.[3]
112.2 The general Hilbert symbol If K is a local field containing the group of nth roots of unity for some positive integer n prime to the characteristic of K, then the Hilbert symbol (,) is a function from K*×K* to μn. In terms of the Artin symbol it can be defined by[4] √ √ √ n n n (a, b) b = (a, K( b)/K) b Hilbert originally defined the Hilbert symbol before the Artin symbol was discovered, and his definition (for n prime) used the power residue symbol when K has residue characteristic coprime to n, and was rather complicated when K has residue characteristic dividing n.
112.2. THE GENERAL HILBERT SYMBOL
112.2.1
369
Properties
The Hilbert symbol is (multiplicatively) bilinear: (ab,c) = (a,c)(b,c) (a,bc) = (a,b)(a,c) skew symmetric: (a,b) = (b,a)−1 nondegenerate: (a,b)=1 for all b if and only if a is in K*n It detects norms (hence the name norm residue symbol): (a,b)=1 if and only if a is a norm of an element in K(n √b) It has the “symbol” properties: (a,1–a)=1, (a,–a)=1.
112.2.2
Hilbert’s reciprocity law
Hilbert’s reciprocity law states that if a and b are in an algebraic number field containing the nth roots of unity then[5] ∏ (a, b)p = 1 p
where the product is over the finite and infinite primes p of the number field, and where (,)p is the Hilbert symbol of the completion at p. Hilbert’s reciprocity law follows from the Artin reciprocity law and the definition of the Hilbert symbol in terms of the Artin symbol.
112.2.3
Power residue symbol
If K is a number field containing the nth roots of unity, p is a prime ideal not dividing n, π is a prime element of the local field of p, and a is coprime to p, then the power residue symbol (a p) is related to the Hilbert symbol by[6] ( ) a = (π, a)p p The power residue symbol is extended to fractional ideals by multiplicativity, and defined for elements of the number field by putting (a b)=(a (b)) where (b) is the principal ideal generated by b. Hilbert’s reciprocity law then implies the following reciprocity law for the residue symbol, for a and b prime to each other and to n: ( ) ( ) ∏ a b = (a, b)p b a p|n,∞
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112.3 External links • Hazewinkel, Michiel, ed. (2001), “Norm-residue symbol”, Encyclopedia of Mathematics, Springer, ISBN 9781-55608-010-4 • HilbertSymbol at Mathworld
112.4 References [1] Lam (2005) pp.450–451 [2] Lam (2005) p.451 [3] Lam (2005) p.455 [4] Neukirch (1999) p.333 [5] Neukirch (1999) p.334 [6] Neukirch (1999) p.336
• Borevich, Z. I.; Shafarevich, I. R. (1966), Number theory, Academic Press, ISBN 0-12-117851-X, Zbl 0145.04902 • Hilbert, David (1897), “Die Theorie der algebraischen Zahlkörper”, Jahresbericht der Deutschen MathematikerVereinigung (in German) 4: 175–546, ISSN 0012-0456 • Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, ISBN 9783-540-62779-1, MR 1646901 • Lam, Tsit-Yuen (2005), Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society, ISBN 0-8218-1095-2, Zbl 1068.11023 • Milnor, John Willard (1971), Introduction to algebraic K-theory, Annals of Mathematics Studies 72, Princeton University Press, MR 0349811, Zbl 0237.18005 • Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021 • Serre, Jean-Pierre (1996), A Course in Arithmetic, Graduate Texts in Mathematics 7, Berlin, New York: Springer-Verlag, ISBN 978-3-540-90040-5, Zbl 0256.12001 • Vostokov, S. V.; Fesenko, I. B. (2002), Local fields and their extensions, Translations of Mathematical Monographs 121, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3259-2, Zbl 1156.11046
Chapter 113
Hilbert’s ninth problem Hilbert’s ninth problem, from the list of 23 Hilbert’s problems (1900), asked to find the most general reciprocity law for the norm residues of k-th order in a general algebraic number field, where k is a power of a prime.
113.1 Progress made The problem was partially solved by Emil Artin (1924; 1927; 1930) by establishing the Artin reciprocity law which deals with abelian extensions of algebraic number fields. Together with the work of Teiji Takagi and Helmut Hasse (who established the more general Hasse reciprocity law), this led to the development of the class field theory, realizing Hilbert’s program in an abstract fashion. Certain explicit formulas for norm residues were later found by Igor Shafarevich (1948; 1949; 1950). The non-abelian generalization, also connected with Hilbert’s twelfth problem, is one of the long-standing challenges in number theory and is far from being complete.
113.2 See also • List of unsolved problems in mathematics
113.3 References • Tate, John (1976). “Problem 9: The general recicprocity law”. In Felix E. Browder. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII.2. American Mathematical Society. pp. 311–322. ISBN 0-8218-1428-1.
113.4 External links • English translation of Hilbert’s original address
371
Chapter 114
Hilbert’s twelfth problem Es handelt sich um meinen liebsten Jugendtraum, nämlich um den Nachweis, dass die Abel ’schen Gleichungen mit Quadratwurzeln rationaler Zahlen durch die Transformations- Gleichungen elliptischer Functionen mit singularen Moduln grade so erschöpft werden, wie die ganzzahligen Abel’schen Gleichungen durch die Kreistheilungsgleichungen. Kronecker in a letter to Dedekind in 1880 reproduced in volume V of his collected works, page 455 Kronecker’s Jugendtraum or Hilbert’s twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic fields and their subfields. The classical theory of complex multiplication, now often known as the Kronecker Jugendtraum, does this for the case of any imaginary quadratic field, by using modular functions and elliptic functions chosen with a particular period lattice related to the field in question. Goro Shimura extended this to CM fields. The general case is still open as of 2014. Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum or “dearest dream of his youth”.
114.1 Description of the problem The fundamental problem of algebraic number theory is to describe the fields of algebraic numbers. The work of Galois made it clear that field extensions are controlled by certain groups, the Galois groups. The simplest situation, which is already at the boundary of what we can do, is when the group in question is abelian. All quadratic extensions, obtained by adjoining the roots of a quadratic polynomial, are abelian, and their study was commenced by Gauss. Another type of abelian extension of the field Q of rational numbers is given by adjoining the nth roots of unity, resulting in the cyclotomic fields. Already Gauss had shown that, in fact, every quadratic field is contained in a larger cyclotomic field. The Kronecker–Weber theorem shows that any finite abelian extension of Q is contained in a cyclotomic field. Kronecker’s (and Hilbert’s) question addresses the situation of a more general algebraic number field K: what are the algebraic numbers necessary to construct all abelian extensions of K? The complete answer to this question has been completely worked out only when K is an imaginary quadratic field or its generalization, a CM-field. Hilbert’s original statement of his 12th problem is rather misleading: he seems to imply that the abelian extensions of imaginary quadratic fields are generated by special values of elliptic modular functions, which is not correct. (It is hard to tell exactly what Hilbert was saying, one problem being that he may have been using the term “elliptic function” to mean both the elliptic function ℘ and the elliptic modular function j.) First it is also necessary to use roots of unity, though Hilbert may have implicitly meant to include these. More seriously, while values of elliptic abelian extensions one also needs to use values of modular functions generate the Hilbert class field, for more general √ elliptic functions. For example, the abelian extension Q(i, 4 1 + 2i)/Q(i) is not generated by singular moduli and roots of unity. One particularly appealing way to state the Kronecker–Weber theorem is by saying that the maximal abelian extension 372
114.2. MODERN DEVELOPMENT
373
of Q can be obtained by adjoining the special values exp(2πi/n) of the exponential function. Similarly, the theory of complex multiplication shows that the maximal abelian extension of Q(τ), where τ is an imaginary quadratic irrationality, can be obtained by adjoining the special values of ℘(τ,z) and j(τ) of modular functions j and elliptic functions ℘, and roots of unity, where τ is in the imaginary quadratic field and z represents a torsion point on the corresponding elliptic curve. One interpretation of Hilbert’s twelfth problem asks to provide a suitable analogue of exponential, elliptic, or modular functions, whose special values would generate the maximal abelian extension Kab of a general number field K. In this form, it remains unsolved. A description of the field Kab was obtained in the class field theory, developed by Hilbert himself, Emil Artin, and others in the first half of the 20th century.[note 1] However the construction of Kab in class field theory involves first constructing larger non-abelian extensions using Kummer theory, and then cutting down to the abelian extensions, so does not really solve Hilbert’s problem which asks for a more direct construction of the abelian extensions.
114.2 Modern development Developments since around 1960 have certainly contributed. Before that Hecke (1912) in his dissertation used Hilbert modular forms to study abelian extensions of real quadratic fields. Complex multiplication of abelian varieties was an area opened up by the work of Shimura and Taniyama. This gives rise to abelian extensions of CM-fields in general. The question of which extensions can be found is that of the Tate modules of such varieties, as Galois representations. Since this is the most accessible case of l-adic cohomology, these representations have been studied in depth. Robert Langlands argued in 1973 that the modern version of the Jugendtraum should deal with Hasse–Weil zeta functions of Shimura varieties. While he envisaged a grandiose program that would take the subject much further, more than thirty years later serious doubts remain concerning its import for the question that Hilbert asked. A separate development was Stark’s conjecture (Harold Stark), which in contrast dealt directly with the question of finding interesting, particular units in number fields. This has seen a large conjectural development for L-functions, and is also capable of producing concrete, numerical results.
114.3 Notes [1] In particular, Teiji Takagi proved the existence of the absolute abelian extension as the well-known Takagi existence theorem.
114.4 References • Langlands, R. P. (1976). “Some contemporary problems with origins in the Jugendtraum”. In Browder, Felix E. Mathematical developments arising from Hilbert problems (PDF). Proc. Sympos. Pure Math. 28. Providence, RI: American Mathematical Society. pp. 401–418. ISBN 0-8218-1428-1. Zbl 0345.14006. • Schappacher, Norbert (1998). “On the history of Hilbert’s twelfth problem: a comedy of errors”. Matériaux pour l'histoire des mathématiques au XXe siècle (Nice, 1996). Sémin. Congr. 3. Paris: Société Mathématique de France. pp. 243–273. ISBN 978-2-85629-065-1. MR 1640262. Zbl 1044.01530. • Vlǎduţ, S. G. (1991). Kronecker’s Jugendtraum and modular functions. Studies in the Development of Modern Mathematics 2. New York: Gordon and Breach Science Publishers. ISBN 2-88124-754-7. Zbl 0731.11001.
Chapter 115
Hurwitz problem In mathematics, the Hurwitz problem, named after Adolf Hurwitz, is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables. There are well-known multiplicative relationships between sums of squares in two variables
(x2 + y 2 )(u2 + v 2 ) = (xu − yv)2 + (xv + yu)2 , (known as the Brahmagupta–Fibonacci identity), and also Euler’s four-square identity and Degen’s eight-square identity. These may be interpreted as muliplicativity for the norms on the complex numbers, quaternions and octonions respectively.[1] The Hurwitz problem for the field K is to find general relations of the form
(x21 + · · · + x2r ) · (y12 + · · · + ys2 ) = (z12 + · · · + zn2 ) , with the z being bilinear forms in the x and y: that is, each z is a K-linear combination of terms of the form xiyj.[2] We call a triple (r, s, n) admissible for K if such an identity exists.[3] Trivial cases of admissible triples include (r, s, rs). The problem is uninteresting for K of characteristic 2, since over such fields every sum of squares is a square, and we exclude this case. It is believed that otherwise admissibility is independent of the field of definition.[4] Hurwitz posed the problem in 1898 in the special case r = s = n and showed that, when coefficients are taken in C, the only admissible values (n, n, n) were n = 1, 2, 4, 8:[5] his proof extends to any field of characteristic not 2.[6] The “Hurwitz–Radon” problem is that of finding admissible triples of the form (r, n, n). Obviously (1, n, n) is admissible. The Hurwitz–Radon theorem states that (ρ(n), n, n) is admissible over any field where ρ(n) is the function defined for n = 2u v, v odd, u = 4a + b, 0 ≤ b ≤ 3, as ρ(n) = 8a + 2b .[5][4] Other admissible triples include (3,5,7)[7] and (10, 10, 16).[4]
115.1 See also • Composition algebra • Hurwitz’s theorem (normed division algebras) • Radon–Hurwitz number
115.2 References [1] Rajwade (1993) pp. 1–3
374
115.2. REFERENCES
375
[2] Lam (2005) p. 127 [3] Rajwade (1993) p. 125 [4] Rajwade (1993) p. 137 [5] Lam (2005) p. 130 [6] Rajwade (1993) p. 3 [7] Rajwade (1993) p. 138
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
Chapter 116
Hyper-finite field In mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field F is called hyper-finite if it is uncountable and quasi-finite, and for every subfield E, every absolutely entire E-algebra (regular field extension of E) of smaller cardinality than F can be embedded in F. They were introduced by Ax (1968). Every hyper-finite field is a pseudo-finite field, and is in particular a model for the first-order theory of finite fields.
116.1 References • Ax, James (1968), “The Elementary Theory of Finite Fields”, Annals of Mathematics, Second Series (Annals of Mathematics) 88 (2): 239–271, doi:10.2307/1970573, ISSN 0003-486X, MR 0229613, Zbl 0195.05701
376
Chapter 117
Hyperreal number "*R” redirects here. For R*, see R* (disambiguation). The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form
1 + 1 + · · · + 1. Such a number is infinite, and its reciprocal is infinitesimal. The term “hyper-real” was introduced by Edwin Hewitt in 1948.[1] The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz’s heuristic Law of Continuity. The transfer principle states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since sin πn = 0 for all integers n, one also has sin πH = 0 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955. Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion.[2] In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules which Robinson delineated. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as derivative and integral in a direct fashion,( without passing ) via logical complications of multiple quantifiers. Thus, the derivative of f (x+∆x)−f (x) ′ f(x) becomes f (x) = st for an infinitesimal ∆x , where st(·) denotes the standard part function, ∆x which “rounds off” each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.
117.1 The transfer principle Main article: Transfer principle The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form “for any number x...” that is true for the reals is also true for the hyperreals. For example, the axiom that states “for any number x, x + 0 = x" still applies. The same is true for quantification over several numbers, e.g., “for any numbers x and y, xy = yx.” This ability to carry over statements from the reals to the hyperreals is called the transfer principle. However, statements of the form “for any set of numbers S ...” may not carry over. The only properties that differ 377
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between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties. The kinds of logical sentences that obey this restriction on quantification are referred to as statements in first-order logic. The transfer principle, however, doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that
1 < ω,
1 + 1 < ω,
1 + 1 + 1 < ω,
1 + 1 + 1 + 1 < ω, . . . .
but there is no such number in R. (In other words, *R is not Archimedean.) This is possible because the nonexistence of ω cannot be expressed as a first order statement.
117.2 Use in analysis 117.2.1
Calculus with algebraic functions
Informal notations for non-real quantities have historically appeared in calculus in two contexts: as infinitesimals like dx and as the symbol ∞, used, for example, in limits of integration of improper integrals. As an example of the transfer principle, the statement that for any nonzero number x, 2x ≠ x, is true for the real numbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers. This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals. Similarly, the casual use of 1/0 = ∞ is invalid, since the transfer principle applies to the statement that division by zero is undefined. The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite. For any finite hyperreal number x, its standard part, st x, is defined as the unique real number that differs from it only infinitesimally. The derivative of a function y(x) is defined not as dy/dx but as the standard part of dy/dx. For example, to find the derivative f′ (x) of the function f(x) = x2 , let dx be a non-zero infinitesimal. Then,
The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the dx2 term. In the hyperreal system, dx2 ≠ 0, since dx is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity dx2 is infinitesimally small compared to dx; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.
117.2.2
Integration
One way of defining a definite integral in the hyperreal system is as the standard part of an infinite sum on a hyperfinite lattice defined as a, a + dx, a + 2dx, ... a + ndx, where dx is infinitesimal, n is an infinite hypernatural, and the lower and upper bounds of integration are a and b = a + n dx.[3]
117.3 Properties The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology. The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Vladimir Kanovei and Shelah[4]
117.4. DEVELOPMENT
379
shows that there is a definable, countably saturated (meaning ω-saturated, but not of course countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the continuum hypothesis. The condition of being a hyperreal field is a stronger one than that of being a real closed field strictly containing R. It is also stronger than that of being a superreal field in the sense of Dales and Woodin.[5]
117.4 Development The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.
117.4.1
From Leibniz to Robinson
When Newton and (more explicitly) Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by George Berkeley. Berkeley’s criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals (or fluxions), where dx is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion (see Ghosts of departed quantities for details). When in the 1800s calculus was put on a firm footing through the development of the (ε, δ)-definition of limit by Bolzano, Cauchy, Weierstrass, and others, infinitesimals were largely abandoned, though research in non-Archimedean fields continued (Ehrlich 2006). However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of non-standard analysis.[6] Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic. Hyper-real fields were in fact originally introduced by Hewitt (1948) by purely algebraic techniques, using an ultrapower construction.
117.4.2
The ultrapower construction
We are going to construct a hyperreal field via sequences of reals.[7] In fact we can add and multiply sequences componentwise; for example:
(a0 , a1 , a2 , . . .) + (b0 , b1 , b2 , . . .) = (a0 + b0 , a1 + b1 , a2 + b2 , . . .) and analogously for multiplication. This turns the set of such sequences into a commutative ring, which is in fact a real algebra A. We have a natural embedding of R in A by identifying the real number r with the sequence (r, r, r, ...) and this identification preserves the corresponding algebraic operations of the reals. The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be self-consistent and well defined. For example, we may have two sequences that differ in their first n members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequences oscillate randomly forever, and we must find some way of taking such a sequence and interpreting it as, say, 7 + ϵ , where ϵ is a certain infinitesimal number. Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion:
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(a0 , a1 , a2 , . . .) ≤ (b0 , b1 , b2 , . . .) ⇐⇒ a0 ≤ b0 ∧ a1 ≤ b1 ∧ a2 ≤ b2 . . . but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. It follows that the relation defined in this way is only a partial order. To get around this, we have to specify which positions matter. Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of index sets that matter is given by any free ultrafilter U on the natural numbers; these can be characterized as ultrafilters which do not contain any finite sets. (The good news is that Zorn’s lemma guarantees the existence of many such U; the bad news is that they cannot be explicitly constructed.) We think of U as singling out those sets of indices that “matter": We write (a0 , a1 , a2 , ...) ≤ (b0 , b1 , b2 , ...) if and only if the set of natural numbers { n : an ≤ bn } is in U. This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a≤b and b≤a. With this identification, the ordered field *R of hyperreals is constructed. From an algebraic point of view, U allows us to define a corresponding maximal ideal I in the commutative ring A (namely, the set of the sequences that vanish in some element of U), and then to define *R as A/I; as the quotient of a commutative ring by a maximal ideal, *R is a field. This is also notated A/U, directly in terms of the free ultrafilter U; the two are equivalent. The maximality of I follows from the possibility of, given a sequence a, constructing a sequence b inverting the non-null elements of a and not altering its null entries. If the set on which a vanishes is not in U, the product ab is identified with the number 1, and any ideal containing 1 must be A. In the resulting field, these a and b are inverses. The field A/U is an ultrapower of R. Since this field contains R it has cardinality at least that of the continuum. Since A has cardinality (2ℵ0 )ℵ0 = 2ℵ0 = 2ℵ0 , 2
it is also no larger than 2ℵ0 , and hence has the same cardinality as R. One question we might ask is whether, if we had chosen a different free ultrafilter V, the quotient field A/U would be isomorphic as an ordered field to A/V. This question turns out to be equivalent to the continuum hypothesis; in ZFC with the continuum hypothesis we can prove this field is unique up to order isomorphism, and in ZFC with the negation of continuum hypothesis we can prove that there are non-order-isomorphic pairs of fields which are both countably indexed ultrapowers of the reals. For more information about this method of construction, see ultraproduct.
117.4.3
An intuitive approach to the ultrapower construction
The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by Goldblatt.[8] Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense; the true infinitesimals include certain classes of sequences that contain a sequence converging to zero. Let us see where these classes come from. Consider first the sequences of real numbers. They form a ring, that is, one can multiply, add and subtract them, but not always divide by a non-zero element. The real numbers are considered as the constant sequences, the sequence is zero if it is identically zero, that is, an = 0 for all n. In our ring of sequences one can get ab = 0 with neither a = 0 nor b = 0. Thus, if for two sequences a, b one has ab = 0, at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As a result, the equivalence classes of sequences that differ by some sequence declared zero will form a field which is called a hyperreal field. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, including those represented by sequences diverging to infinity). Also every hyperreal which is not infinitely large will be infinitely close to an ordinary real, in other words, it will be the sum of an ordinary real and an infinitesimal. This construction is parallel to the construction of the reals from the rationals given by Cantor. He started with the ring of the Cauchy sequences of rationals and declared all the sequences that converge to zero to be zero. The result is the reals. To continue the construction of hyperreals, let us consider the zero sets of our sequences, that is, the z(a) = {i : ai = 0} , that is, z(a) is the set of indexes i for which ai = 0 . It is clear that if ab = 0 , then the union of z(a) and z(b) is N (the set of all natural numbers), so:
117.5. PROPERTIES OF INFINITESIMAL AND INFINITE NUMBERS
381
1. One of the sequences that vanish on two complementary sets should be declared zero 2. If a
is declared zero, ab
3. If both a
and b
should be declared zero too, no matter what b
are declared zero, then a2 + b2
is.
should also be declared zero.
Now the idea is to single out a bunch U of subsets X of N and to declare that a = 0 to U. From the above conditions one can see that:
if and only if z(a)
belongs
1. From two complementary sets one belongs to U 2. Any set containing a set that belongs to U, also belongs to U. 3. An intersection of any two sets belonging to U belongs to U. 4. Finally, we do not want an empty set to belong to U because then everything becomes zero, as every set contains an empty set. Any family of sets that satisfies (2–4) is called a filter (an example: the complements to the finite sets, it is called the Fréchet filter and it is used in the usual limit theory). If (1) also holds, U is called an ultrafilter (because you can add no more sets to it without breaking it). The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10). Such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers. Any ultrafilter containing a finite set is trivial. It is known that any filter can be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added as an extra axiom, as it is weaker than the axiom of choice. Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter) and do our construction, we get the hyperreal numbers as a result. If f is a real function of a real variable x variable by composition:
then f
naturally extends to a hyperreal function of a hyperreal
f ({xn }) = {f (xn )} where {. . . } means “the equivalence class of the sequence . . . relative to our ultrafilter”, two sequences being in the same class if and only if the zero set of their difference belongs to our ultrafilter. All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals. One can prove that any finite (that is, such that |x| < a for some ordinary real a ) hyperreal x will be of the form y + d where y is an ordinary (called standard) real and d is an infinitesimal. Now one can see that f is continuous means that f (a) − f (x) is differentiable means that
is infinitely small whenever x − a
is, and f
(f (x) − f (a))/(x − a) − f ′ (a) is infinitely small whenever x − a is. Remarkably, if one allows a matically continuous (because, f being differentiable at x ,
to be hyperreal, the derivative will be auto-
f ′ (x) − (f (x) − f (a))/(x − a) = f ′ (x) − (f (a) − f (x))/(a − x) is infinitely small when x − a
is, therefore f ′ (x) − f ′ (a)
is also infinitely small when x − a
is).
117.5 Properties of infinitesimal and infinite numbers The finite elements F of *R form a local ring, and in fact a valuation ring, with the unique maximal ideal S being the infinitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from
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F to R whose kernel consists of the infinitesimals and which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal. Put another way, every finite nonstandard real number is “very close” to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x, conceptually the same as x to the nearest real number. This operation is an order-preserving homomorphism and hence is well-behaved both algebraically and order theoretically. It is order-preserving though not isotonic; i.e. x ≤ y implies st(x) ≤ st(y) , but x < y does not imply st(x) < st(y) . • We have, if both x and y are finite,
st(x + y) = st(x) + st(y) st(xy) = st(x) st(y) • If x is finite and not infinitesimal.
st(1/x) = 1/ st(x) • x is real if and only if
st(x) = x The map st is continuous with respect to the order topology on the finite hyperreals; in fact it is locally constant.
117.6 Hyperreal fields Suppose X is a Tychonoff space, also called a T₃.₅ space, and C(X) is the algebra of continuous real-valued functions on X. Suppose M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered field F containing the reals. If F strictly contains R then M is called a hyperreal ideal (terminology due to Hewitt (1948)) and F a hyperreal field. Note that no assumption is being made that the cardinality of F is greater than R; it can in fact have the same cardinality. An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra Rκ of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.
117.7 See also • Hyperinteger • Real closed fields • Non-standard calculus • Constructive non-standard analysis • Influence of non-standard analysis • Surreal number
117.8. REFERENCES
383
117.8 References [1] Hewitt (1948), p. 74, as reported in Keisler (1994) [2] Ball, p. 31 [3] Keisler [4] Kanovei, Vladimir; Shelah, Saharon (2004), “A definable nonstandard model of the reals” (PDF), Journal of Symbolic Logic 69: 159–164, doi:10.2178/jsl/1080938834 [5] Woodin, W. H.; Dales, H. G. (1996), Super-real fields: totally ordered fields with additional structure, Oxford: Clarendon Press, ISBN 978-0-19-853991-9 [6] Robinson, Abraham (1996), Non-standard analysis, Princeton University Press, ISBN 978-0-691-04490-3. The classic introduction to nonstandard analysis. [7] Loeb, Peter A. (2000), “An introduction to nonstandard analysis”, Nonstandard analysis for the working mathematician, Math. Appl. 510, Dordrecht: Kluwer Acad. Publ., pp. 1–95 [8] Goldblatt, Robert (1998), Lectures on the hyperreals: an introduction to nonstandard analysis, Berlin, New York: SpringerVerlag, ISBN 978-0-387-98464-3
117.9 Further reading • Ball, W.W. Rouse (1960), A Short Account of the History of Mathematics (4th ed. [Reprint. Original publication: London: Macmillan & Co., 1908] ed.), New York: Dover Publications, pp. 50–62, ISBN 0-486-20630-0 • Hatcher, William S. (1982) “Calculus is Algebra”, American Mathematical Monthly 89: 362–370. • Hewitt, Edwin (1948) Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64, 45—99. • Jerison, Meyer; Gillman, Leonard (1976), Rings of continuous functions, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90198-5 • Keisler, H. Jerome (1994) The hyperreal line. Real numbers, generalizations of the reals, and theories of continua, 207—237, Synthese Lib., 242, Kluwer Acad. Publ., Dordrecht. • Kleinberg, Eugene M.; Henle, James M. (2003), Infinitesimal Calculus, New York: Dover Publications, ISBN 978-0-486-42886-4
117.10 External links • Crowell, Calculus. A text using infinitesimals. • Hermoso, Nonstandard Analysis and the Hyperreals. A gentle introduction. • Keisler, Elementary Calculus: An Approach Using Infinitesimals. Includes an axiomatic treatment of the hyperreals, and is freely available under a Creative Commons license • Stroyan, A Brief Introduction to Infinitesimal Calculus Lecture 1 Lecture 2 Lecture 3
Chapter 118
Ideal class group In mathematics, for a field K an ideal class group (or class group) is the quotient group JK/PK where JK is the whole fractional ideals of K and PK is the principal ideals of K. The extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by the ideal class group (or class group). If this group is finite (as it is in the case of the ring of integers of a number field), then the order of the group is called the class number. The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.
118.1 History and origin of the ideal class group Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Gauss, a composition law was defined on certain equivalence classes of forms. This gave a finite abelian group, as was recognised at the time. Later Kummer was working towards a theory of cyclotomic fields. It had been realised (probably by several people) that failure to complete proofs in the general case of Fermat’s last theorem by factorisation using the roots of unity was for a very good reason: a failure of the fundamental theorem of arithmetic to hold in the rings generated by those roots of unity was a major obstacle. Out of Kummer’s work for the first time came a study of the obstruction to the factorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion in that group for the field of p-roots of unity, for any prime number p, as the reason for the failure of the standard method of attack on the Fermat problem (see regular prime). Somewhat later again Dedekind formulated the concept of ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integers is a Dedekind domain). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal domain; a ring is a principal domain if and only if it has a trivial ideal class group.
118.2 Definition If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that (a)I = (b)J. (Here the notation (a) means the principal ideal of R consisting of all the multiples of a.) It is easily shown that this is an equivalence relation. The equivalence classes are called the ideal classes of R. Ideal classes can be multiplied: if [I] denotes the equivalence class of the ideal I, then the multiplication [I][J] = [IJ] is well-defined and commutative. The principal ideals form the ideal class [R] which serves as an identity element for this multiplication. Thus a class [I] has an inverse [J] if and only if there is an ideal J such that IJ is a principal ideal. In general, such a J may not exist and consequently the set of ideal classes of R may only be a monoid. However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, 384
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the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except R) is a product of prime ideals.
118.3 Properties The ideal class group is trivial (i.e. has only one element) if and only if all ideals of R are principal. In this sense, the ideal class group measures how far R is from being a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains). The number of ideal classes (the class number of R) may be infinite in general. In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain.[1] But if R is in fact a ring of algebraic integers, then the class number is always finite. This is one of the main results of classical algebraic number theory. Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an algebraic number field of small discriminant, using Minkowski’s bound. This result gives a bound, depending on the ring, such that every ideal class contains an ideal norm less than the bound. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task. The mapping from rings of integers R to their corresponding class groups is functorial, and the class group can be subsumed under the heading of algebraic K-theory, with K 0 (R) being the functor assigning to R its ideal class group; more precisely, K 0 (R) = Z×C(R), where C(R) is the class group. Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers.
118.4 Relation with the group of units It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a Dedekind domain behave like elements. The other part of the answer is provided by the multiplicative group of units of the Dedekind domain, since passage from principal ideals to their generators requires the use of units (and this is the rest of the reason for introducing the concept of fractional ideal, as well): Define a map from K × to the set of all nonzero fractional ideals of R by sending every element to the principal (fractional) ideal it generates. This is a group homomorphism; its kernel is the group of units of R, and its cokernel is the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.
118.5 Examples of ideal class groups • The rings Z, Z[ω], and Z[i], where ω is a cube root of 1 and i is a fourth root of 1 (i.e. a square root of −1), are all principal ideal domains (and in fact are all Euclidean domains), and so have class number 1: that is, they have trivial ideal class groups. • If k is a field, then the polynomial ring k[X1 , X2 , X3 , ...] is an integral domain. It has a countably infinite set of ideal classes.
118.5.1
Class numbers of quadratic fields
If d is a square-free integer (a product of distinct primes) other than 1, then Q(√d) is a quadratic extension of Q. If d < 0, then the class number of the ring R of algebraic integers of Q(√d) is equal to 1 for precisely the following values of d: d = −1, −2, −3, −7, −11, −19, −43, −67, and −163. This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner’s proof was not believed until Harold Stark gave a later proof in 1967. (See Stark-Heegner theorem.) This is a special case of the famous class number problem. If, on the other hand, d > 0, then it is unknown whether there are infinitely many fields Q(√d) with class number 1. Computational results indicate that there are a great many such fields. However, it is not even known if there are infinitely many number fields with class number 1.[2][3]
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For d < 0, the ideal class group of Q(√d) is isomorphic to the class group of integral binary quadratic forms of discriminant equal to the discriminant of Q(√d). For d > 0, the ideal class group may be half the size since the class group of integral binary quadratic forms is isomorphic to the narrow class group of Q(√d).[4] Example of a non-trivial class group The quadratic integer ring R = Z [√−5] is the ring of integers of Q(√−5). It does not possess unique factorization; in fact the class group of R is cyclic of order 2. Indeed, the ideal J = (2, 1 + √−5) √ is not principal, which can be proved by contradiction as follows. R has a norm function N (a + b −5) = a2 + 5b2 , which satisfies N (uv) = N (u)N (v) , and N√(u) = 1 if and only if u is a unit in R . First of all, J ̸= R , because the quotient ring of R modulo the ideal (1 + −5) is isomorphic to Z/6Z , so that the quotient ring of R modulo J is isomorphic to Z/2Z . If J were generated by an element √ x of R, then x would divide both 2 and 1 + √−5. Then the norm N (x) would divide both N (2) = 4 and N (1 + 5) = 6 , so N(x) would divide 2. If N (x) = 1 , then x is a unit, and J = R , a contradiction. But N (x) cannot be 2 either, because R has no elements of norm 2, because the Diophantine equation b2 + 5c2 = 2 has no solutions in integers, as it has no solutions modulo 5. One also computes that J 2 = (2), which is principal, so the class of J in the ideal class group has order two. Showing that there aren't any other ideal classes requires more effort. The fact that this J is not principal is also related to the fact that the element 6 has two distinct factorisations into irreducibles: 6 = 2 × 3 = (1 + √−5) × (1 − √−5).
118.6 Connections to class field theory Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extension of such a field. The Hilbert class field L of a number field K is unique and has the following properties: • Every ideal of the ring of integers of K becomes principal in L, i.e., if I is an integral ideal of K then the image of I is a principal ideal in L. • L is a Galois extension of K with Galois group isomorphic to the ideal class group of K. Neither property is particularly easy to prove.
118.7 See also • Class number formula • Class number problem • Brauer–Siegel theorem—an asymptotic formula for the class number • List of number fields with class number one • Principal ideal domain • Algebraic K-theory • Galois theory • Fermat’s last theorem
118.8. NOTES
387
• Narrow class group • Picard group—a generalisation of the class group appearing in algebraic geometry • Arakelov class group
118.8 Notes [1] Claborn 1966 [2] Neukirch 1999 [3] Gauss 1700 [4] Fröhlich & Taylor 1993, Theorem 58
118.9 References • Claborn, Luther (1966), “Every abelian group is a class group”, Pacific Journal of Mathematics 18: 219–222, doi:10.2140/pjm.1966.18.219 • Fröhlich, Albrecht; Taylor, Martin (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, ISBN 978-0-521-43834-6, MR 1215934 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
Chapter 119
Ideal norm In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.
119.1 Relative norm Let A be a Dedekind domain with field of fractions K and integral closure of B in a finite separable extension L of K. (this implies that B is also a Dedekind domain.) Let IA and IB be the ideal groups of A and B, respectively (i.e., the sets of nonzero fractional ideals.) Following (Serre 1979), the norm map
NB/A : IB → IA is the unique group homomorphism that satisfies
NB/A (q) = p[B/q:A/p] for all nonzero prime ideals q of B, where p = q ∩ A is the prime ideal of A lying below q . Alternatively, for any b ∈ IB one can equivalently define NB/A (b) to be the fractional ideal of A generated by the set {NL/K (x)|x ∈ b} of field norms of elements of B.[1] For a ∈ IA , one has NB/A (aB) = an , where n = [L : K] . The ideal norm of a principal ideal is thus compatible with the field norm of an element: NB/A (xB) = NL/K (x)A. [2] Let L/K be a Galois extension of number fields with rings of integers OK ⊂ OL . Then the preceding applies with A = OK , B = OL , and for any b ∈ IOL we have
NOL /OK (b) = OK ∩
∏
σ(b),
σ∈Gal(L/K)
which is an element of IOK . The notation NOL /OK is sometimes shortened to NL/K , an abuse of notation that is compatible with also writing NL/K for the field norm, as noted above. In the case K = Q , it is reasonable to use positive rational numbers as the range for NOL /Z since Z has trivial ideal class group and unit group {±1} , thus each nonzero fractional ideal of Z is generated by a uniquely determined positive rational number. Under this convention the relative norm from L down to K = Q coincides with the absolute norm defined below. 388
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119.2 Absolute norm Let L be a number field with ring of integers OL , and a a nonzero (integral) ideal of OL . The absolute norm of a is
N (a) := [OL : a] = |OL /a|. By convention, the norm of the zero ideal is taken to be zero. If a = (a) is a principal ideal, then N (a) = |NL/Q (a)| .[3] The norm is completely multiplicative: if a and b are ideals of OL , then N (a · b) = N (a)N (b) .[4] Thus the absolute norm extends uniquely to a group homomorphism
N : IOL → Q× >0 , defined for all nonzero fractional ideals of OL . The norm of an ideal a can be used to give an upper bound on the field norm of the smallest nonzero element it contains: there always exists a nonzero a ∈ a for which ( )s √ 2 |NL/Q (a)| ≤ |∆L |N (a), π where ∆L is the discriminant of L and s is the number of pairs of (non-real) complex embeddings of L into C (the number of complex places of L ).[5]
119.3 See also • Field norm • Dedekind zeta function
119.4 References [1] Janusz, Proposition I.8.2 [2] Serre, 1.5, Proposition 14. [3] Marcus, Theorem 22c, pp. 65-66. [4] Marcus, Theorem 22a, pp. 65-66 [5] Neukirch, Lemma 6.2
• Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics 7 (second ed.), Providence, Rhode Island: American Mathematical Society, pp. x+276, ISBN 0-8218-0429-4, MR 1362545 (96j:11137) • Marcus, Daniel A. (1977), Number fields, Universitext, New York: Springer-Verlag, pp. viii+279, ISBN 0387-90279-1, MR 0457396 (56 #15601) • Jürgen Neukirch (1999), Algebraic number theory, Berlin: Springer-Verlag, pp. xviii+571, ISBN 3-54065399-6, MR 1697859 (2000m:11104) • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Translated from the French by Marvin Jay Greenberg, New York: Springer-Verlag, pp. viii+241, ISBN 0-387-90424-7, MR 554237 (82e:12016)
Chapter 120
Integer This article is about numbers traditionally known as “integers”. For computer representations, see integer (computer science). For the concept in algebraic number theory, see integral element. An integer (from the Latin integer meaning “whole”)[note 1] is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5½, and √2 are not. The set of integers consists of zero (0), the natural numbers (1, 2, 3, …), also called whole numbers or counting numbers,[1] and their additive inverses (the negative integers, i.e. −1, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold Z (Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], “numbers”).[2][3] ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the (rational) integers are the algebraic integers that are also rational numbers.
120.1 Algebraic properties
Integers can be thought of as discrete, equally spaced points on an infinitely long number line. In the above, non-negative integers are shown in purple and negative integers in red.
Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, 0, Z (unlike the natural numbers) is also closed under subtraction. The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring Z. Z is not closed under division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative). The following lists some of the basic properties of addition and multiplication for any integers a, b and c. In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every non-zero integer can be written as a finite sum 1 + 1 + … + 1 or (−1) + (−1) + … + (−1). In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the left hand side is even, while the right hand side is odd. This means that Z under multiplication is not a group. 390
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All the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in Z for all values of variables, which are true in any unital commutative ring. Note that certain non-zero integers map to zero in certain rings. At last, the property (*) says that the commutative ring Z is an integral domain. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field with the usual operations containing the integers is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes Z as its subring. Although ordinary division is not defined on Z, the division “with remainder” is defined on them. It is called Euclidean division and possesses the following important property: that is, given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < | b |, where | b | denotes the absolute value of b. The integer q is called the quotient and r is called the remainder of the division of a by b. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.
120.2 Order-theoretic properties Z is a totally ordered set without upper or lower bound. The ordering of Z is given by: … −3 < −2 < −1 < 0 < 1 < 2 < 3 < … An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: 1. if a < b and c < d, then a + c < b + d 2. if a < b and 0 < c, then ac < bc. It follows that Z together with the above ordering is an ordered ring. The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.[4] This is equivalent to the statement that any Noetherian valuation ring is either a field or a discrete valuation ring.
120.3 Construction In elementary school teaching, integers are often intuitively defined as the (positive) natural numbers, zero, and the negations of the natural numbers. However, this style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that these operations obey the laws of arithmetic.[5] Therefore, in modern set-theoretic mathematics a more abstract construction,[6] which allows one to define the arithmetical operations without any case distinction, is often used instead.[7] The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b).[8] The intuition is that (a,b) stands for the result of subtracting b from a.[8] To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:
(a, b) ∼ (c, d)
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Red points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line.
precisely when
a + d = b + c. Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;[8] denoting by [(a,b)] the equivalence class having (a,b) as a member, one has:
[(a, b)] + [(c, d)] := [(a + c, b + d)]. [(a, b)] · [(c, d)] := [(ac + bd, ad + bc)]. The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
−[(a, b)] := [(b, a)]. Hence subtraction can be defined as the addition of the additive inverse:
120.4. COMPUTER SCIENCE
393
[(a, b)] − [(c, d)] := [(a + d, b + c)]. The standard ordering on the integers is given by: [(a, b)] < [(c, d)] iff a + d < b + c. It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes. Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). The natural number n is identified with the class [(n,0)] (in other words the natural numbers are embedded into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted −n (this covers all remaining classes, and gives the class [(0,0)] a second time since −0 = 0. Thus, [(a,b)] is denoted by {
a − b, if a ≥ b −(b − a), if a < b.
If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity. This notation recovers the familiar representation of the integers as {…, −2, −1, 0, 1, 2, …}. Some examples are: 0 = [(0, 0)] 1 = [(1, 0)]
= [(1, 1)] = · · · = [(2, 1)] = · · ·
= [(k, k)] = [(k + 1, k)]
−1 = [(0, 1)] 2 = [(2, 0)] −2 = [(0, 2)]
= [(1, 2)] = · · · = [(3, 1)] = · · · = [(1, 3)] = · · ·
= [(k, k + 1)] = [(k + 2, k)] = [(k, k + 2)].
120.4 Computer science Main article: Integer (computer science) An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity. Also, in the common two’s complement representation, the inherent definition of sign distinguishes between “negative” and “non-negative” rather than “negative, positive, and 0”. (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.). Variable-length representations of integers, such as bignums, can store any integer that fits in the computer’s memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).
120.5 Cardinality The cardinality of the set of integers is equal to ℵ0 (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from Z to N. If N = {0, 1, 2, …} then consider the function: { 2|x|, if x ≤ 0 f (x) = 2x − 1, if x > 0.
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{… (−4,8) (−3,6) (−2,4) (−1,2) (0,0) (1,1) (2,3) (3,5) …} If N = {1, 2, 3, ...} then consider the function: { 2|x|, if x < 0 g(x) = 2x + 1, if x ≥ 0. {… (−4,8) (−3,6) (−2,4) (−1,2) (0,1) (1,3) (2,5) (3,7) …} If the domain is restricted to Z then each and every member of Z has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality.
120.6 See also • 0.999... • Canonical representation of a positive integer • Hyperinteger • Integer-valued function • Integer lattice • Integer part • Integer sequence • Profinite integer
120.7 Notes [1] Integer 's first, literal meaning in Latin is “untouched”, from in (“not”) plus tangere (“to touch”). "Entire" derives from the same origin via French (see: Evans, Nick (1995). “A-Quantifiers and Scope”. In Bach, Emmon W. Quantification in Natural Languages. Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262. ISBN 0-7923-33527.)
120.8 References [1] Weisstein, Eric W., “Counting Number”, and “Whole Number”, MathWorld. [2] Miller, Jeff (2010-08-29). “Earliest Uses of Symbols of Number Theory”. Retrieved 2010-09-20. [3] Peter Jephson Cameron (1998). Introduction to Algebra. Oxford University Press. p. 4. ISBN 978-0-19-850195-4. [4] Warner, Seth (2012), Modern Algebra, Dover Books on Mathematics, Courier Corporation, Theorem 20.14, p. 185, ISBN 9780486137094. [5] Mendelson, Elliott (2008), Number Systems and the Foundations of Analysis, Dover Books on Mathematics, Courier Dover Publications, p. 86, ISBN 9780486457925. [6] Ivorra Castillo: Álgebra [7] Frobisher, Len (1999), Learning to Teach Number: A Handbook for Students and Teachers in the Primary School, The Stanley Thornes Teaching Primary Maths Series, Nelson Thornes, p. 126, ISBN 9780748735150. [8] Campbell, Howard E. (1970). The structure of arithmetic. Appleton-Century-Crofts. p. 83. ISBN 0-390-16895-5.
120.9. SOURCES
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120.9 Sources • Bell, E.T., Men of Mathematics. New York: Simon and Schuster, 1986. (Hardcover; ISBN 0-671-464000)/(Paperback; ISBN 0-671-62818-6) • Herstein, I.N., Topics in Algebra, Wiley; 2 edition (June 20, 1975), ISBN 0-471-01090-1. • Mac Lane, Saunders, and Garrett Birkhoff; Algebra, American Mathematical Society; 3rd edition (April 1999). ISBN 0-8218-1646-2. • Weisstein, Eric W., “Integer”, MathWorld.
120.10 External links • Hazewinkel, Michiel, ed. (2001), “Integer”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • The Positive Integers — divisor tables and numeral representation tools • On-Line Encyclopedia of Integer Sequences cf OEIS This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
Chapter 121
Isomorphism extension theorem In field theory, a branch of mathematics, the isomorphism extension theorem is an important theorem regarding the extension of a field isomorphism to a larger field.
121.1 Isomorphism extension theorem The theorem states that given any field F , an algebraic extension field E of F and an isomorphism ϕ mapping F onto a field F ′ then ϕ can be extended to an isomorphism τ mapping E onto an algebraic extension E ′ of F ′ (a subfield of the algebraic closure of F ′ ). The proof of the isomorphism extension theorem depends on Zorn’s lemma.
121.2 References • D.J. Lewis, Introduction to algebra, Harper & Row, 1965, Chap.IV.12, p.193.
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Chapter 122
Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Iwasawa (1959), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 90s), Ralph Greenberg has proposed an Iwasawa theory for motives.
122.1 Formulation Iwasawa worked with so-called Zp -extensions: infinite extensions of a number field F with Galois group Γ isomorphic n to the additive group of p-adic integers for some prime p. Every closed subgroup of Γ is of the form Γp , so by Galois theory, a Zp -extension F∞ /F is the same thing as a tower of fields F = F0 ⊂ F1 ⊂ F2 ⊂ . . . ⊂ F∞ such that Gal(Fn /F ) ∼ = Z/pn Z . Iwasawa studied classical Galois modules over Fn by asking questions about the structure of modules over F∞ . More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.
122.2 Example Let p be a prime number and let K = Q(μp) be the field generated over Q by the pth roots of unity. Iwasawa considered the following tower of number fields:
K = K0 ⊂ K1 ⊂ · · · ⊂ K∞ , ∪ where Kn is the field generated by adjoining to K the pn +1st roots of unity and K∞ = Kn . The fact that Gal(Kn /K) ≃ Z/pn Z implies, by infinite Galois theory, that Gal(K∞ /K) is isomorphic to Zp . In order to get an interesting Galois module here, Iwasawa took the ideal class group of Kn , and let In be its p-torsion part. There are norm maps Im → In whenever m > n , and this gives us the data of an inverse system. If we set I = lim In , ←− then it is not hard to see from the inverse limit construction that I is a module over Zp . In fact, I is a module over the Iwasawa algebra Λ = Zp [[Γ]] . This is a 2-dimensional, regular local ring, and this makes it possible to describe modules over it. From this description it is possible to recover information about the p-part of the class group of K . The motivation here is that the p-torsion in the ideal class group of K had already been identified by Kummer as the main obstruction to the direct proof of Fermat’s last theorem.
122.3 Connections with p-adic analysis From this beginning in the 1950s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The 397
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latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory had prospects of moving ahead finally from Kummer’s century-old results on regular primes. Iwasawa formulated the main conjecture of Iwasawa theory as an assertion that two methods of defining p-adic Lfunctions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by Mazur & Wiles (1984) for Q, and for all totally real number fields by Wiles (1990). These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand’s theorem (the so-called Herbrand-Ribet theorem). Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin’s Euler systems, described in Lang (1990) and Washington (1997), and later proved other generalizations of the main conjecture for imaginary quadratic fields.
122.4 Generalizations The Galois group of the infinite tower, the starting field, and the sort of arithmetic module studied can all be varied. In each case, there is a main conjecture linking the tower to a p-adic L-function. In 2002, Chris Skinner and Eric Urban claimed a proof of a main conjecture for GL(2). In 2010, they posted a preprint (Skinner & Urban 2010).
122.5 See also • Ferrero–Washington theorem • Tate module of a number field
122.6 References • Coates, J.; Sujatha, R. (2006), Cyclotomic Fields and Zeta Values, Springer Monographs in Mathematics, Springer-Verlag, ISBN 3-540-33068-2, Zbl 1100.11002 • Greenberg, Ralph (2001), “Iwasawa theory---past and present”, in Miyake, Katsuya, Class field theory---its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math. 30, Tokyo: Math. Soc. Japan, pp. 335–385, ISBN 978-4-931469-11-2, MR 1846466, Zbl 0998.11054 • Iwasawa, Kenkichi (1959), “On Γ-extensions of algebraic number fields”, Bulletin of the American Mathematical Society 65 (4): 183–226, doi:10.1090/S0002-9904-1959-10317-7, ISSN 0002-9904, MR 0124316, Zbl 0089.02402 • Kato, Kazuya (2007), “Iwasawa theory and generalizations”, in Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis et al., International Congress of Mathematicians. Vol. I (PDF), Eur. Math. Soc., Zürich, pp. 335–357, doi:10.4171/022-1/14, ISBN 978-3-03719-022-7, MR 2334196 • Lang, Serge (1990), Cyclotomic fields I and II, Graduate Texts in Mathematics 121, With an appendix by Karl Rubin (Combined 2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96671-7, Zbl 0704.11038 • Mazur, Barry; Wiles, Andrew (1984), “Class fields of abelian extensions of Q", Inventiones Mathematicae 76 (2): 179–330, doi:10.1007/BF01388599, ISSN 0020-9910, MR 742853, Zbl 0545.12005 • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, MR 2392026, Zbl 1136.11001 • Rubin, Karl (1991), “The ‘main conjectures’ of Iwasawa theory for imaginary quadratic fields”, Inventiones Mathematicae 103 (1): 25–68, doi:10.1007/BF01239508, ISSN 0020-9910, Zbl 0737.11030 • Skinner, Chris; Urban, Éric (2010), The Iwasawa main conjectures for GL2 (PDF), p. 219
122.7. FURTHER READING
399
• Washington, Lawrence C. (1997), Introduction to cyclotomic fields, Graduate Texts in Mathematics 83 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4 • Andrew Wiles (1990), “The Iwasawa Conjecture for Totally Real Fields”, Annals of Mathematics (Annals of Mathematics) 131 (3): 493–540, doi:10.2307/1971468, JSTOR 1971468, Zbl 0719.11071.
122.7 Further reading • de Shalit, Ehud (1987), Iwasawa theory of elliptic curves with complex multiplication. p-adic L functions, Perspectives in Mathematics 3, Boston etc.: Academic Press, ISBN 0-12-210255-X, Zbl 0674.12004
122.8 External links • Hazewinkel, Michiel, ed. (2001), “Iwasawa theory”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4
Chapter 123
Jacobson–Bourbaki theorem In algebra, the Jacobson–Bourbaki theorem is a theorem used to extend Galois theory to field extensions that need not be separable. It was introduced by Nathan Jacobson (1944) for commutative fields and extended to noncommutative fields by Jacobson (1947), and Cartan (1947) who credited the result to unpublished work by Nicolas Bourbaki. The extension of Galois theory to normal extensions is called the Jacobson–Bourbaki correspondence, which replaces the correspondence between some subfields of a field and some subgroups of a Galois group by a correspondence between some sub division rings of a division ring and some subalgebras of an algebra. The Jacobson–Bourbaki theorem implies both the usual Galois correspondence for subfields of a Galois extension, and Jacobson’s Galois correspondence for subfields of a purely inseparable extension of exponent at most 1.
123.1 Statement Suppose that L is a division ring. The Jacobson–Bourbaki theorem states that there is a natural 1:1 correspondence between: • Division rings K in L of finite index n (in other words L is a finite-dimensional left vector space over K). • Unital K-algebras of finite dimension n (as K-vector spaces) contained in the ring of endomorphisms of the additive group of K. The sub division ring and the corresponding subalgebra are each other’s commutants. Jacobson (1956, Chapter 7.2) gave an extension to sub division rings that might have infinite index, which correspond to closed subalgebras in the finite topology.
123.2 References • Cartan, Henri (1947), “Les principaux théorèmes de la théorie de Galois pour les corps non nécessairement commutatifs”, Les Comptes rendus de l'Académie des sciences 224: 249–251, MR 0020983 • Cartan, Henri (1947), “Théorie de Galois pour les corps non commutatifs”, Annales Scientifiques de l'École Normale Supérieure. Troisième Série 64: 59–77, ISSN 0012-9593, MR 0023237 • Jacobson, Nathan (1944), “Galois theory of purely inseparable fields of exponent one”, American Journal of Mathematics 66: 645–648, doi:10.2307/2371772, ISSN 0002-9327, MR R0011079 • Jacobson, Nathan (1947), “A note on division rings”, American Journal of Mathematics 69: 27–36, doi:10.2307/2371651, ISSN 0002-9327, MR 0020981 • Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications 37, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1037-8, MR 0081264 400
123.2. REFERENCES
401
• Jacobson, Nathan (1964), Lectures in abstract algebra. Vol III: Theory of fields and Galois theory, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London-New York, ISBN 978-0-387-90168-8, MR 0172871 • Kreimer, F. (2001), “Jacobson-Bourbaki_theorem”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 124
Krasner’s lemma In number theory, more specifically in p-adic analysis, Krasner’s lemma is a basic result relating the topology of a complete non-archimedean field to its algebraic extensions.
124.1 Statement Let K be a complete non-archimedean field and let K be a separable closure of K. Given an element α in K, denote its Galois conjugates by α2 , ..., αn. Krasner’s lemma states:[1][2] if an element β of K is such that |α − β| < |α − αi | for i = 2, . . . , n then K(α) ⊆ K(β).
124.2 Applications • Krasner’s lemma can be used to show that p -adic completion and separable closure of global fields commute.[3] In other words, given p a prime of a global field L, the separable closure of the p -adic completion of L equals the p -adic completion of the separable closure of L (where p is a prime of L above p ). • Another application is to proving that C — the completion of the algebraic closure of Q — is algebraically closed.[4][5]
124.3 Generalization Krasner’s lemma has the following generalization.[6] Consider a monic polynomial
f∗ =
n ∏
(X − αk∗ )
k=1
of degree n > 1 with coefficients in a Henselian field (K, v) and roots in the algebraic closure K. Let I and J be two disjoint, non-empty sets with union {1,...,n}. Moreover, consider a polynomial
g=
∏ (X − αi ) i∈I
402
124.4. NOTES
403
with coefficients and roots in K. Assume
∀i ∈ I∀j ∈ J : v(αi − αi∗ ) > v(αi∗ − αj∗ ). Then the coefficients of the polynomials
g ∗ :=
∏ ∏ (X − αi∗ ), h∗ := (X − αj∗ ) i∈I
j∈J
are contained in the field extension of K generated by the coefficients of g. (The original Krasner’s lemma corresponds to the situation where g has degree 1.)
124.4 Notes [1] Lemma 8.1.6 of Neukirch, Schmidt & Wingberg 2008 [2] Lorenz (2008) p.78 [3] Proposition 8.1.5 of Neukirch, Schmidt & Wingberg 2008 [4] Proposition 10.3.2 of Neukirch, Schmidt & Wingberg 2008 [5] Lorenz (2008) p.80 [6] Brink (2006), Theorem 6
124.5 References
• Brink, David (2006). “New light on Hensel’s Lemma”. Expositiones Mathematicae 24 (4): 291–306. doi:10.1016/j.exmath.2006.0 ISSN 0723-0869. Zbl 1142.12304. • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. SpringerVerlag. ISBN 978-0-387-72487-4. Zbl 1130.12001. • Narkiewicz, Władysław (2004). Elementary and analytic theory of algebraic numbers. Springer Monographs in Mathematics (3rd ed.). Berlin: Springer-Verlag. p. 206. ISBN 3-540-21902-1. Zbl 1159.11039. • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, Zbl 1136.11001, MR 2392026
Chapter 125
Kronecker–Weber theorem In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q. The Kronecker–Weber theorem provides a partial converse: every abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients. For example, √ 5 = e2πi/5 − e4πi/5 − e6πi/5 + e8πi/5 . The theorem is named after Leopold Kronecker and Heinrich Martin Weber.
125.1 Field-theoretic formulation The Kronecker–Weber theorem can be stated in terms of fields and field extensions. Precisely, the Kronecker-Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field. That is, whenever an algebraic number field has a Galois group over Q that is an abelian group, the field is a subfield of a field obtained by adjoining a root of unity to the rational numbers. For a given abelian extension K of Q there is a minimal cyclotomic field that contains it. The theorem allows one to define the conductor of K as the smallest integer n such that K lies inside the field generated by the n-th roots of unity. For example the quadratic fields have as conductor the absolute value of their discriminant, a fact generalised in class field theory.
125.2 History The theorem was first stated by Kronecker (1853) though his argument was not complete for extensions of degree a power of 2. Weber (1886) published a proof, but this had some gaps and errors that were pointed out and corrected by Neumann (1981). The first complete proof was given by Hilbert (1896).
125.3 Generalizations Lubin and Tate (1965, 1966) proved the local Kronecker–Weber theorem which states that any abelian extension of a local field can be constructed using cyclotomic extensions and Lubin–Tate extensions. Hazewinkel (1975), Rosen (1981) and Lubin (1981) gave other proofs. Hilbert’s twelfth problem asks for generalizations of the Kronecker–Weber theorem to base fields other than the rational numbers, and asks for the analogues of the roots of unity for those fields. 404
125.4. REFERENCES
405
125.4 References • Ghate, Eknath (2000), “The Kronecker-Weber theorem”, in Adhikari, S. D.; Katre, S. A.; Thakur, Dinesh, Cyclotomic fields and related topics (Pune, 1999) (PDF), Bhaskaracharya Pratishthana, Pune, pp. 135–146, MR 1802379 • Greenberg, M. J. (1974). “An Elementary Proof of the Kronecker-Weber Theorem”. American Mathematical Monthly (The American Mathematical Monthly, Vol. 81, No. 6) 81 (6): 601–607. doi:10.2307/2319208. JSTOR 2319208. • Hazewinkel, Michiel (1975), “Local class field theory is easy”, Advances in Mathematics 18 (2): 148–181, doi:10.1016/0001-8708(75)90156-5, ISSN 0001-8708, MR 0389858 • Hilbert, David (1896), “Ein neuer Beweis des Kronecker’schen Fundamentalsatzes über Abel’sche Zahlkörper.”, Nachrichten der Gesellschaft der Wissenschaften zu Göttingen (in German): 29–39 • Kronecker, Leopold (1853), "Über die algebraisch auflösbaren Gleichungen”, Berlin K. Akad. Wiss. (in German): 365–374, Collected works volume 4 • Kronecker, Leopold (1877), "Über Abelsche Gleichungen”, Berlin K. Akad. Wiss. (in German): 845–851, Collected works volume 4 • Lemmermeyer, Franz (2005), “Kronecker-Weber via Stickelberger”, Journal de théorie des nombres de Bordeaux 17 (2): 555–558, doi:10.5802/jtnb.507, ISSN 1246-7405, MR 2211307 • Lubin, Jonathan (1981), “The local Kronecker-Weber theorem”, Transactions of the American Mathematical Society 267 (1): 133–138, doi:10.2307/1998574, ISSN 0002-9947, MR 621978 • Lubin, Jonathan; Tate, John (1965), “Formal complex multiplication in local fields”, Annals of Mathematics. Second Series 81: 380–387, ISSN 0003-486X, JSTOR 1970622, MR 0172878 • Lubin, Jonathan; Tate, John (1966), “Formal moduli for one-parameter formal Lie groups”, Bulletin de la Société Mathématique de France 94: 49–59, ISSN 0037-9484, MR 0238854 • Neumann, Olaf (1981), “Two proofs of the Kronecker-Weber theorem “according to Kronecker, and Weber"", Journal für die reine und angewandte Mathematik 323: 105–126, doi:10.1515/crll.1981.323.105, ISSN 00754102, MR 611446 • Rosen, Michael (1981), “An elementary proof of the local Kronecker-Weber theorem”, Transactions of the American Mathematical Society 265 (2): 599–605, doi:10.2307/1999753, ISSN 0002-9947, MR 610968 • Šafarevič, I. R. (1951), A new proof of the Kronecker-Weber theorem, Trudy Mat. Inst. Steklov. (in Russian) 38, Moscow: Izdat. Akad. Nauk SSSR, pp. 382–387, MR 0049233 English translation in his Collected Mathematical Papers • Schappacher, Norbert (1998), “On the history of Hilbert’s twelfth problem: a comedy of errors”, Matériaux pour l'histoire des mathématiques au XXe siècle (Nice, 1996), Sémin. Congr. 3, Paris: Société Mathématique de France, pp. 243–273, ISBN 978-2-85629-065-1, MR 1640262 • Weber, H. (1886), “Theorie der Abel’schen Zahlkörper”, Acta Mathematica (in German) 8: 193–263, doi:10.1007/BF02417089, ISSN 0001-5962
Chapter 126
Kummer theory In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat’s last theorem. The main statements do not depend on the nature of the field - apart from its characteristic, which should not divide the integer n – and therefore belong to abstract algebra. The theory of cyclic extensions of the field K when the characteristic of K does divide n is called Artin–Schreier theory. Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots. The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is something much more serious.
126.1 Kummer extensions A Kummer extension is a field extension L/K, where for some given integer n > 1 we have • K contains n distinct nth roots of unity (i.e., roots of Xn −1) • L/K has abelian Galois group of exponent n. For example, when n = 2, the first condition is always true if K has characteristic ≠ 2. The Kummer extensions in this case include quadratic extensions L = K(√a) where a in K is a non-square element. By the usual solution of quadratic equations, any extension of degree 2 of K has this form. The Kummer extensions in this case also include biquadratic extensions and more general multiquadratic extensions. When K has characteristic 2, there are no such Kummer extensions. Taking n = 3, there are no degree 3 Kummer extensions of the rational number field Q, since for three cube roots of 1 complex numbers are required. If one takes L to be the splitting field of X3 − a over Q, where a is not a cube in the rational numbers, then L contains a subfield K with three cube roots of 1; that is because if α and β are roots of the cubic polynomial, we shall have (α/β)3 =1 and the cubic is a separable polynomial. Then L/K is a Kummer extension. More generally, it is true that when K contains n distinct nth roots of unity, which implies that the characteristic of K doesn't divide n, then adjoining to K the nth root of any element a of K creates a Kummer extension (of degree m, for some m dividing n). As the splitting field of the polynomial Xn − a, the Kummer extension is necessarily Galois, √ with Galois group that is cyclic of order m. It is easy to track the Galois action via the root of unity in front of n a. Kummer theory provides converse statements. When K contains n distinct nth roots of unity, it states that any abelian extension of K of exponent dividing n is formed by extraction of roots of elements of K. Further, if K × denotes the multiplicative group of non-zero elements of K, abelian extensions of K of exponent n correspond bijectively with subgroups of K × /(K × )n , 406
126.2. GENERALIZATIONS
407
that is, elements of K × modulo nth powers. The correspondence can be described explicitly as follows. Given a subgroup
∆ ⊆ K × /(K × )n , the corresponding extension is given by
K(∆1/n ), √ where ∆1/n = { n a : a ∈ K × , a · (K × )n ∈ ∆} . In fact it suffices to adjoin nth root of one representative of each element of Δ. Conversely, if L is a Kummer extension of K, then Δ is recovered by the rule
∆ = (K × ∩ (L× )n )/(K × )n . In this case there is an isomorphism
∆∼ = Homc (Gal(L/K), µn ) given by ( a 7→
) σ(α) σ 7→ , α
where α is any nth root of a in L. Here µn denotes the multiplicative group of nth roots of unity (which belong to K) and Homc (Gal(L/K), µn ) is the group of continuous homomorphisms from Gal(L/K) equipped with Krull topology to µn with discrete topology (with group operation given by pointwise multiplication). This group (with discrete topology) can also be viewed as Pontryagin dual of Gal(L/K) , assuming we regard µn as a subgroup of circle group. If the extension L/K is finite, then Gal(L/K) is a finite discrete group and we have
∆∼ = Hom(Gal(L/K), µn ) ∼ = Gal(L/K), however the last isomorphism isn't natural.
126.2 Generalizations Suppose that G is a profinite group acting on a module A with a surjective homomorphism π from the G-module A to itself. Suppose also that G acts trivially on the kernel C of π and that the first cohomology group H1 (G,A) is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between AG /π(AG ) and Hom(G,C). Kummer theory is the special case of this when A is the multiplicative group of the separable closure of a field k, G is the Galois group, π is the nth power map, and C the group of nth roots of unity. Artin–Schreier theory is the special case when A is the additive group of the separable closure of a field k of positive characteristic p, G is the Galois group, π is the Frobenius map, and C the finite field of order p. Taking A to be a ring of truncated Witt vectors gives Witt’s generalization of Artin–Schreier theory to extensions of exponent dividing pn .
126.3 See also • Quadratic field
408
CHAPTER 126. KUMMER THEORY
126.4 References • Hazewinkel, Michiel, ed. (2001), “Kummer extension”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4 • Bryan Birch, “Cyclotomic fields and Kummer extensions”, in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.III, pp. 85–93.
Chapter 127
Lafforgue’s theorem In mathematics, Lafforgue’s theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups and representations of Galois groups. The Langlands conjectures were introduced by Langlands (1967, 1970) and describe a correspondence between representations of the Weil group of an algebraic function field and representations of algebraic groups over the function field, generalizing class field theory of function fields from abelian Galois groups to non-abelian Galois groups.
127.1 Langlands conjectures for GL1 The Langlands conjectures for GL1 (K) follow from (and are essentially equivalent to) class field theory. More precisely the Artin map gives a map from the idele class group to the abelianization of the Weil group.
127.2 Representations of the Weil group 127.3 Automorphic representations of GLn(F) The representations of GLn(F) appearing in the Langlands correspondence are automorphic representations.
127.4 Drinfeld’s theorem for GL2 127.5 Lafforgue’s theorem for GLn(F) Here F is a global field of some positive characteristic p, and ℓ is some prime not equal to p. Lafforgue’s theorem states that there is a bijection σ between: • Equivalence classes of cuspidal representations π of GLn(F), and • Equivalence classes of irreducible ℓ-adic representations σ(π) of dimension n of the absolute Galois group of F that preserves the L-function at every place of F. The proof of Lafforgue’s theorem involves constructing a representation σ(π) of the absolute Galois group for each cuspidal representation π. The idea of doing this is to look in the ℓ-adic cohomology of the moduli stack of shtukas of rank n that have compatible level N structures for all N. The cohomology contains subquotients of the form 409
410
CHAPTER 127. LAFFORGUE’S THEOREM π⊗σ(π)⊗σ(π)∨
which can be used to construct σ(π) from π. A major problem is that the moduli stack is not of finite type, which means that there are formidable technical difficulties in studying its cohomology.
127.6 Applications Lafforgue’s theorem implies the Ramanujan–Petersson conjecture that if an automorphic form for GLn(F) has central character of finite order, then the corresponding Hecke eigenvalues at every unramified place have absolute value 1. Lafforgue’s theorem implies the conjecture of Deligne (1980, 1.2.10) that an irreducible finite-dimensional l-adic representation of the absolute Galois group with determinant character of finite order is pure of weight 0.
127.7 See also • Local Langlands conjectures
127.8 References • Borel, Armand (1979), “Automorphic L-functions”, in Borel, Armand; Casselman, W., Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, ISBN 978-0-8218-1437-6, MR 546608 • Deligne, Pierre (1980), “La conjecture de Weil. II”, Publications Mathématiques de l'IHÉS (52): 137–252, ISSN 1618-1913, MR 601520 • Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) [1966], Representation theory and automorphic functions, Generalized functions 6, Philadelphia, Pa.: W. B. Saunders Co., ISBN 978-0-12-279506-0, MR 0220673 • Lafforgue, Laurent (1998), “Proceedings of the International Congress of Mathematicians (Berlin, 1998)", Documenta Mathematica II: 563–570, ISSN 1431-0635, MR 1648105 |chapter= ignored (help) • Lafforgue, Laurent Chtoucas de Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands. (Drinfeld shtukas, Arthur-Selberg trace formula and Langlands correspondence) Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 383–400, Higher Ed. Press, Beijing, 2002. • Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, MR 0401654 • Langlands, Robert (1967), Letter to Prof. Weil • Langlands, R. P. (1970), “Problems in the theory of automorphic forms”, Lectures in modern analysis and applications, III, Lecture Notes in Math 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614 • Gérard Laumon, The work of Laurent Lafforgue Proceedings of the ICM, Beijing 2002, vol. 1, 91–97 • G. Laumon La correspondance de Langlands sur les corps de fonctions (d'apres Laurent Lafforgue) (The Langlands correspondence over function fields (according to Laurent Lafforgue)) Seminaire Bourbaki, 52eme annee, 1999–2000, no. 873
127.9. EXTERNAL LINKS
127.9 External links • Lafforgue’s publications • The work of Robert Langlands • Rapoport, The work of Laurent Lafforgue
411
Chapter 128
Langlands dual group Not to be confused with Langlands group. In representation theory, a branch of mathematics, the Langlands dual L G of a reductive algebraic group G (also called the L-group of G) is a group that controls the representation theory of G. If G is defined over a field k, then L G is an extension of the absolute Galois group of k by a complex Lie group. There is also a variation called the Weil form of the L-group, where the Galois group is replaced by a Weil group. The Langlands dual group is also often referred to as an L-group; here the letter L indicates also the connection with the theory of L-functions, particularly the automorphic L-functions. The Langlands dual was introduced by Langlands (1967) in a letter to A. Weil. The L-group is used heavily in the Langlands conjectures of Robert Langlands. It is used to make precise statements from ideas that automorphic forms are in a sense functorial in the group G, when k is a global field. It is not exactly G with respect to which automorphic forms and representations are functorial, but L G. This makes sense of numerous phenomena, such as 'lifting' of forms from one group to another larger one, and the general fact that certain groups that become isomorphic after field extensions have related automorphic representations.
128.1 Definition for separably closed fields From a reductive algebraic group over a separably closed field K we can construct its root datum (X* , Δ,X*, Δv ), where X* is the lattice of characters of a maximal torus, X* the dual lattice (given by the 1-parameter subgroups), Δ the roots, and Δv the coroots. A connected reductive algebraic group over K is uniquely determined (up to isomorphism) by its root datum. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group. For any root datum (X* , Δ,X*, Δv ), we can define a dual root datum (X*, Δv ,X* , Δ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots. If G is a connected reductive algebraic group over the algebraically closed field K, then its Langlands dual group L G is the complex connected reductive group whose root datum is dual to that of G. Examples: The Langlands dual group L G has the same Dynkin diagram as G, except that components of type Bn are changed to components of type Cn and vice versa. If G has trivial center then L G is simply connected, and if G is simply connected then L G has trivial center. The Langlands dual of GLn(K) is GLn(C).
128.2 Definition for groups over more general fields Now suppose that G is a reductive group over some field k with separable closure K. Over K, G has a root datum, and this comes with an action of the Galois group Gal(K/k). The identity component L Go of the L-group is the connected complex reductive group of the dual root datum; this has an induced action of the Galois group Gal(K/k). The full L-group L G is the semidirect product L
G = L Go ×Gal(K/k) 412
128.3. APPLICATIONS
413
of the connected component with the Galois group. There are some variations of the definition of the L-group, as follows: • Instead of using the full Galois group Gal(K/k) of the separable closure, one can just use the Galois group of a finite extension over which G is split. The corresponding semidirect product then has only a finite number of components and is a complex Lie group. • Suppose that k is a local, global, or finite field. Instead of using the absolute Galois group of k, one can use the absolute Weil group, which has a natural map to the Galois group and therefore also acts on the root datum. The corresponding semidirect product is called the Weil form of the L-group. • For algebraic groups G over finite fields, Deligne and Lusztig introduced a different dual group. As before, G gives a root datum with an action of the absolute Galois group of the finite field. The dual group G* is then the reductive algebraic group over the finite field associated to the dual root datum with the induced action of the Galois group. (This dual group is defined over a finite field, while the component of the Langlands dual group is defined over the complex numbers.)
128.3 Applications The Langlands conjectures imply, very roughly, that if G is a reductive algebraic group over a local or global field, then there is a correspondence between “good” representations of G and homomorphisms of a Galois group (or Weil group or Langlands group) into the Langlands dual group of G. A more general formulation of the conjectures is Langlands functoriality, which says (roughly) that given a (well behaved) homomorphism between Langlands dual groups, there should be an induced map between “good” representations of the corresponding groups. To make this theory explicit, there must be defined the concept of L-homomorphism of an L-group into another. That is, L-groups must be made into a category, so that 'functoriality' has meaning. The definition on the complex Lie groups is as expected, but L-homomorphisms must be 'over' the Weil group.
128.4 References • A. Borel, Automorphic L-functions, in Automorphic forms, representations, and L-functions, ISBN 0-82181437-0 • Langlands, R. (1967), letter to A. Weil
• Mirković, I.; Vilonen, K. (2007), “Geometric Langlands duality and representations of algebraic groups over commutative rings”, Annals of Mathematics. Second Series 166 (1): 95–143, arXiv:math/0401222, doi:10.4007/annals.2007.166.9 ISSN 0003-486X, MR 2342692 describes the dual group of G in terms of the geometry of the affine Grassmannian of G.
Chapter 129
Langlands–Deligne local constant In mathematics, the Langlands–Deligne local constant (or local Artin root number up to an elementary function of s) is an elementary function associated with a representation of the Weil group of a local field. The functional equation L(ρ,s) = ε(ρ,s)L(ρ∨ ,1−s) of an Artin L-function has an elementary function ε(ρ,s) appearing in it, equal to a constant called the Artin root number times an elementary real function of s, and Langlands discovered that ε(ρ,s) can be written in a canonical way as a product ε(ρ,s) = Π ε(ρv, s, ψv) of local constants ε(ρv, s, ψv) associated to primes v. Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate’s thesis. Dwork (1956) proved the existence of the local constant ε(ρv, s, ψv) up to sign. The original proof of the existence of the local constants by Langlands (1970) used local methods and was rather long and complicated, and never published. Deligne (1973) later discovered a simpler proof using global methods.
129.1 Properties The local constants ε(ρ, s, ψE) depend on a representation ρ of the Weil group and a choice of character ψE of the additive group of E. They satisfy the following conditions: • If ρ is 1-dimensional then ε(ρ, s, ψE) is the constant associated to it by Tate’s thesis as the constant in the functional equation of the local L-function. • ε(ρ1 ⊕ρ2 , s, ψE) = ε(ρ1 , s, ψE)ε(ρ2 , s, ψE). As a result ε(ρ, s, ψE) can also be defined for virtual representations ρ. • If ρ is a virtual representation of dimension 0 and E contains K then ε(ρ, s, ψE) = ε(IndE/Kρ, s, ψK) Brauer’s theorem on induced characters implies that these three properties characterize the local constants. Deligne (1976) showed that the local constants are trivial for real (orthogonal) representations of the Weil group.
129.2 Notational conventions There are several different conventions for denoting the local constants. 414
129.3. REFERENCES
415
• The parameter s is redundant and can be combined with the representation ρ, because ε(ρ, s, ψE) = ε(ρ⊗||s , 0, ψE) for a suitable character ||. • Deligne includes an extra parameter dx consisting of a choice of Haar measure on the local field. Other conventions omit this parameter by fixing a choice of Haar measure: either the Haar measure that is self dual with respect to ψ (used by Langlands), or the Haar measure that gives the integers of E measure 1. These different conventions differ by elementary terms that are positive real numbers.
129.3 References • Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 335, Berlin, New York: SpringerVerlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120, ISBN 978-3-540-31486-8 • Deligne, Pierre (1973), “Les constantes des équations fonctionnelles des fonctions L”, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture notes in mathematics 349, Berlin, New York: Springer-Verlag, pp. 501–597, doi:10.1007/978-3-540-37855-6_7, MR 0349635 • Deligne, Pierre (1976), “Les constantes locales de l'équation fonctionnelle de la fonction L d'Artin d'une représentation orthogonale”, Inventiones Mathematicae 35: 299–316, doi:10.1007/BF01390143, ISSN 00209910, MR 0506172 • Dwork, Bernard (1956), “On the Artin root number”, American Journal of Mathematics 78: 444–472, ISSN 0002-9327, JSTOR 2372524, MR 0082476 • Langlands, Robert (1970), On the functional equation of the Artin L-functions, Unpublished notes • Tate, John T. (1977), “Local constants”, in Fröhlich, A., Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Boston, MA: Academic Press, pp. 89–131, ISBN 978-0-12-268960-4, MR 0457408 • Tate, J. (1979), “Number theoretic background”, Automorphic forms, representations, and L-functions Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 0-8218-1435-4
129.4 External links • Perlis, R. (2001), “Artin root numbers”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 130
Lazard’s universal ring In mathematics, Lazard’s universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let F(x, y) be x + y + Σci,j xi yj for indeterminates ci,j, and we define the universal ring R to be the commutative ring generated by the elements ci,j, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring R has the following universal property: For every commutative ring S, one-dimensional formal group laws over S correspond to ring homomorphisms from R to S. The commutative ring R constructed above is known as Lazard’s universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degrees 2, 4, 6, … (where ci,j has degree 2(i + j − 1)). Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard’s universal ring.
130.1 References • Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 9780-226-00524-9 • Lazard, Michel (1955), “Sur les groupes de Lie formels à un paramètre”, Bulletin de la Société Mathématique de France 83: 251–274, ISSN 0037-9484, MR 0073925 • Lazard, Michel (1975), Commutative formal groups, Lecture Notes in Mathematics 443, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070554, ISBN 978-3-540-07145-7, MR 0393050 • Quillen, Daniel (1969), “On the formal group laws of unoriented and complex cobordism theory”, Bulletin of the American Mathematical Society 75: 1293–1298, doi:10.1090/S0002-9904-1969-12401-8, MR 0253350
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Chapter 131
Leopoldt’s conjecture In algebraic number theory, Leopoldt’s conjecture, introduced by H.-W. Leopoldt (1962, 1975), states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual logarithms, introduced by H.-W. Leopoldt (1962). Leopoldt proposed a definition of a p-adic regulator Rp attached to K and a prime number p. The definition of Rp uses an appropriate determinant with entries the p-adic logarithm of a generating set of units of K (up to torsion), in the manner of the usual regulator. The conjecture, which for general K is still open as of 2009, then comes out as the statement that Rp is not zero.
131.1 Formulation Let K be a number field and for each prime P of K above some fixed rational prime p, let UP denote the local units at P and let U₁,P denote the subgroup of principal units in UP. Set
U1 =
∏
U1,P .
P |p
Then let E 1 denote the set of global units ε that map to U 1 via the diagonal embedding of the global units in E. Since E1 is a finite-index subgroup of the global units, it is an abelian group of rank r1 + r2 − 1 , where r1 is the number of real embeddings of K and r2 the number of pairs of complex embeddings. Leopoldt’s conjecture states that the Zp -module rank of the closure of E1 embedded diagonally in U1 is also r1 + r2 − 1. Leopoldt’s conjecture is known in the special case where K is an abelian extension of Q or an abelian extension of an imaginary quadratic number field: Ax (1965) reduced the abelian case to a p-adic version of Baker’s theorem, which was proved shortly afterwards by Brumer (1967). Mihăilescu (2009, 2011) has announced a proof of Leopoldt’s conjecture for all CM-extensions of Q . Colmez (1988) expressed the residue of the p-adic Dedekind zeta function of a totally real field at s = 1 in terms of the p-adic regulator. As a consequence, Leopoldt’s conjecture for those fields is equivalent to their p-adic Dedekind zeta functions having a simple pole at s = 1.
131.2 References • Ax, James (1965), “On the units of an algebraic number field”, Illinois Journal of Mathematics 9: 584–589, ISSN 0019-2082, MR 0181630, Zbl 0132.28303 • Brumer, Armand (1967), “On the units of algebraic number fields”, Mathematika. A Journal of Pure and Applied Mathematics 14 (2): 121–124, doi:10.1112/S0025579300003703, ISSN 0025-5793, MR 0220694, Zbl 0171.01105 417
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• Colmez, Pierre (1988), “Résidu en s=1 des fonctions zêta p-adiques”, Inventiones Mathematicae 91 (2): 371– 389, doi:10.1007/BF01389373, ISSN 0020-9910, MR 922806, Zbl 0651.12010 • Kolster, M. (2001), “l/l110120”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781-55608-010-4 • Leopoldt, Heinrich-Wolfgang (1962), “Zur Arithmetik in abelschen Zahlkörpern”, Journal für die reine und angewandte Mathematik 209: 54–71, ISSN 0075-4102, MR 0139602, Zbl 0204.07101 • Leopoldt, H. W. (1975), "Eine p-adische Theorie der Zetawerte II", Journal für die reine und angewandte Mathematik 1975 (274/275): 224–239, doi:10.1515/crll.1975.274-275.224, Zbl 0309.12009. • Mihăilescu, Preda (2009), The T and T* components of Λ - modules and Leopoldt’s conjecture, arXiv:0905.1274 • Mihăilescu, Preda (2011), Leopoldt’s Conjecture for CM fields, arXiv:1105.4544 • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, Zbl 1136.11001, MR 2392026 • Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields (Second ed.), New York: Springer, ISBN 0-387-94762-0, Zbl 0966.11047.
Chapter 132
Levi-Civita field In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Its members can be constructed as formal series of the form ∑
aq εq ,
q∈Q
where aq are real numbers, Q is the set of rational numbers, and ε is to be interpreted as a positive infinitesimal. The support of a, i.e., the set of indices of the nonvanishing coefficients {q ∈ Q : aq ̸= 0}, must be a left-finite set: for any member of Q , there are only finitely many members of the set less than it; this restriction is necessary in order to make multiplication and division well defined and unique. The ordering is defined according to dictionary ordering of the list of coefficients, which is equivalent to the assumption that ε is an infinitesimal. The real numbers are embedded in this field as series in which all of the coefficients vanish except a0 .
132.1 Examples • 7ε is an infinitesimal that is greater than ε , but less than every positive real number. • ε2 is less than ε , and is also less than rε for any positive real r . • 1 + ε differs infinitesimally from 1. 1
• ε 2 is greater than ε , but still less than every positive real number. • 1/ε is greater than any real number. • 1 + ε + 12 ε2 + · · · +
1 n n! ε
+ · · · is interpreted as eε .
• 1 + ε + 2ε2 + · · · + n!εn + · · · is a valid member of the field, because the series is to be construed formally, without any consideration of convergence.
132.2 Extensions and applications The field can be algebraically closed by adjoining an imaginary unit (i), or by letting the coefficients be complex. It is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented using floating point. It is the basis of automatic differentiation, a way to perform differentiation in cases that are intractable by symbolic differentiation or finite-difference methods.[1] Hahn series (with real coefficients and value group Q ) are a larger field which relaxes the condition on the support {q ∈ Q : aq ̸= 0} of being left finite to that of being well-ordered (i.e., admitting no infinite decreasing sequence): this gives a meaning to series such as 1 + ε1/2 + ε2/3 + ε3/4 + ε4/5 + · · · which are not in the Levi-Civita field. 419
420
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132.3 References [1] Khodr Shamseddine, “Analysis on the Levi-Civita Field: A Brief Overview”, http://www.physics.umanitoba.ca/~{}khodr/ Publications/RS-Overview-offprints.pdf
132.4 External links • A web-based calculator for Levi-Civita numbers
Chapter 133
Linked field In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.
133.1 Linked quaternion algebras Let F be a field of characteristic not equal to 2. Let A = (a1 ,a2 ) and B = (b1 ,b2 ) be quaternion algebras over F. The algebras A and B are linked quaternion algebras over F if there is x in F such that A is equivalent to (x,y) and B is equivalent to (x,z).[1] The Albert form for A, B is
q = ⟨−a1 , −a2 , a1 a2 , b1 , b2 , −b1 b2 ⟩ . It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B.[2] The quaternion algebras are linked if and only if the Albert form is isotropic.[3]
133.2 Linked fields The field F is linked if any two quaternion algebras over F are linked.[4] Every global and local field is linked since all quadratic forms of dgree 6 over such fields are isotropic. The following properties of F are equivalent:[5] • F is linked. • Any two quaternion algebras over F are linked. • Every Albert form (dimension six form of discriminant −1) is isotropic. • The quaternion algebras form a subgroup of the Brauer group of F. • Every dimension five form over F is a Pfister neighbour. • No biquaternion algebra over F is a division algebra. A nonreal linked field has u-invariant equal to 1,2,4 or 8.[6]
133.3 References [1] Lam (2005) p.69
421
422
CHAPTER 133. LINKED FIELD
[2] Knus, Max-Albert (1991). Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften 294. Berlin etc.: Springer-Verlag. p. 192. ISBN 3-540-52117-8. Zbl 0756.11008. [3] Lam (2005) p.70 [4] Lam (2005) p.370 [5] Lam (2005) p.342 [6] Lam (2005) p.406
133.4 Bibliography • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
133.5 Further reading • Gentile, Enzo R. (1989). “On linked fields”. Rev. Unión Mat. Argent. 35: 67–81. ISSN 0041-6932. Zbl 0823.11010.
Chapter 134
Liouville’s theorem (differential algebra) In mathematics, Liouville’s theorem, originally formulated by Joseph Liouville in the 1830s and 1840s, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A stan2 dard example of such a function is e−x , whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions sin(x) and xx . x Liouville’s theorem states that elementary antiderivatives, if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms.
134.1 Definitions For any differential field F, there is a subfield Con(F) = {f in F | Df = 0}, called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is a simple transcendental extension of F (i.e. G = F(t) for some transcendental t) such that Dt = Ds/s for some s in F. This has the form of a logarithmic derivative. Intuitively, one may think of t as the logarithm of some element s of F, in which case, this condition is analogous to the ordinary chain rule. But it must be remembered that F is not necessarily equipped with a unique logarithm; one might adjoin many “logarithm-like” extensions to F. Similarly, an exponential extension is a simple transcendental extension that satisfies Dt = t Ds. With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, G is called an elementary differential extension of F if there is a finite chain of subfields from F to G where each extension in the chain is either algebraic, logarithmic, or exponential.
134.2 Basic theorem Suppose F and G are differential fields, with Con(F) = Con(G), and that G is an elementary differential extension of F. Let a be in F, y in G, and suppose Dy = a (in words, suppose that G contains an antiderivative of a). Then there exist c1 , ..., cn in Con(F), u1 , ..., un, v in F such that
a = c1
Dun Du1 + · · · + cn + Dv. u1 un 423
424
CHAPTER 134. LIOUVILLE’S THEOREM (DIFFERENTIAL ALGEBRA)
In other words, the only functions that have “elementary antiderivatives” (i.e. antiderivatives living in, at worst, an elementary differential extension of F) are those with this form prescribed by the theorem. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the “simple” functions plus a finite number of logarithms of “simple” functions. A proof of Liouville’s theorem can be found in section 12.4 of Geddes, et al.
134.3 Examples As an example, the field C(x) of rational functions in a single variable has a derivation given by the standard derivative with respect to that variable. The constants of this field are just the complex numbers C. The function x1 , which exists in C(x), does not have an antiderivative in C(x). Its antiderivatives ln x + C do, however, exist in the logarithmic extension C(x, ln x). Likewise, the function x21+1 does not have an antiderivative in C(x). Its antiderivatives tan−1 (x) + C do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of rational functions and logarithms of rational functions. However, a calculation with Euler’s formula shows that in fact the antiderivatives can be written in the required manner (as logarithms of rational functions).
eiθ = cos θ + i sin θ e−iθ = cos θ − i sin θ eiθ cos θ + i sin θ = e−iθ cos θ − i sin θ 1 + i tan θ = 1 − i tan θ
e2iθ =
2iθ = ln 2i tan−1 x = ln tan−1 x =
1 + i tan θ 1 − i tan θ 1 + ix 1 − ix
1 1 + ix ln 2i 1 − ix
134.4 Relationship with differential Galois theory Liouville’s theorem is sometimes presented as a theorem in differential Galois theory, but this is not strictly true. The theorem can be proved without any use of Galois theory. Furthermore, the Galois group of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (corresponding to the constant of integration). Thus, an antiderivative’s differential Galois group does not encode enough information to determine if it can be expressed using elementary functions, the major condition of Liouville’s theorem.
134.5 References • Bertrand, D. (1996), “Review of “Lectures on differential Galois theory"", Bulletin of the American Mathematical Society 33 (2), doi:10.1090/s0273-0979-96-00652-0, ISSN 0002-9904 • Geddes, Czapor, Labahn (1992). Algorithms for Computer Algebra. Kluwer Academic Publishers. ISBN 0-7923-9259-0. • Magid, Andy R. (1994), Lectures on differential Galois theory, University Lecture Series 7, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-7004-4, MR 1301076
134.6. SEE ALSO
425
• Magid, Andy R. (1999), “Differential Galois theory”, Notices of the American Mathematical Society 46 (9): 1041–1049, ISSN 0002-9920, MR 1710665 • van der Put, Marius; Singer, Michael F. (2003), Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 328, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44228-8, MR 1960772
134.6 See also • Risch algorithm
Chapter 135
List of algebraic number theory topics This is a list of algebraic number theory topics.
135.1 Basic topics These topics are basic to the field, either as prototypical examples, or as basic objects of study. • Algebraic number field • Gaussian integer, Gaussian rational • Quadratic field • Cyclotomic field • Cubic field • Biquadratic field • Quadratic reciprocity • Ideal class group • Dirichlet’s unit theorem • Discriminant of an algebraic number field • Ramification (mathematics) • Root of unity • Gaussian period
135.2 Important problems • Fermat’s last theorem • Class number problem for imaginary quadratic fields • Stark–Heegner theorem • Heegner number • Langlands program 426
135.3. GENERAL ASPECTS
135.3 General aspects • Different ideal • Dedekind domain • Splitting of prime ideals in Galois extensions • Decomposition group • Inertia group • Frobenius automorphism • Chebotarev’s density theorem • Totally real field • Local field • p-adic number • p-adic analysis • Adele ring • Idele group • Idele class group • Adelic algebraic group • Global field • Hasse principle • Hasse–Minkowski theorem • Galois module • Galois cohomology • Brauer group
135.4 Class field theory • Class field theory • Abelian extension • Kronecker–Weber theorem • Hilbert class field • Takagi existence theorem • Hasse norm theorem • Artin reciprocity • Local class field theory
427
428
CHAPTER 135. LIST OF ALGEBRAIC NUMBER THEORY TOPICS
135.5 Iwasawa theory • Iwasawa theory • Herbrand–Ribet theorem • Vandiver’s conjecture • Stickelberger’s theorem • Euler system • p-adic L-function
135.6 Arithmetic geometry • Arithmetic geometry • Complex multiplication • Abelian variety of CM-type • Chowla–Selberg formula • Hasse–Weil zeta function
Chapter 136
List of number fields with class number one This is an incomplete list of number fields with class number 1. It is believed that there are infinitely many such number fields, but this has not been proven.[1]
136.1 Definition The class number of a number field is by definition the order of the ideal class group of its ring of integers. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.
136.2
Quadratic number fields
These are of the form K = Q(√d), for a square-free integer d.
136.2.1 Real quadratic fields K is called real quadratic if d > 0. K has class number 1 for the following values of d (sequence A003172 in OEIS): • 2*, 3, 5*, 6, 7, 11, 13*, 14, 17*, 19, 21, 22, 23, 29*, 31, 33, 37*, 38, 41*, 43, 46, 47, 53*, 57, 59, 61*, 62, 67, 69, 71, 73*, 77, 83, 86, 89*, 93, 94, 97*, ...[1][2] (complete until d = 100) *: The narrow class number is also 1 (see related sequence A003655 in OEIS).
Despite what would appear to be the case for these small values, not all prime numbers that are congruent to 1 modulo 4 appear on this list, notably the fields Q(√d) for d = 229 and d = 257 both have class number greater than 1 (in fact equal to 3 in both cases).[3] The density of such primes for which Q(√d) does have class number 1 is conjectured to be nonzero, and in fact close to 76%,[4] however it is not even known whether there are infinitely many real quadratic fields with class number 1.[1]
136.2.2 Imaginary quadratic fields K has class number 1 exactly for the following negative values of d: • −1, −2, −3, −7, −11, −19, −43, −67, −163.[1] 429
430
CHAPTER 136. LIST OF NUMBER FIELDS WITH CLASS NUMBER ONE
(By definition, these also all have narrow class number 1.)
136.3
Cubic fields
The first 60 totally real cubic fields (ordered by discriminant) have class number one. In other words, all cubic fields of discriminant between 0 and 1944 (inclusively) have class number one. The next totally real cubic field (of discriminant 1957) has class number two. The discriminants less than 500 with class number one are: • 49, 81, 148, 169, 229, 257, 316, 321, 361, 404, 469, 473. Polynomials defining the first three are respectively: • x3 − x2 − 2x + 1, • x3 − 3x − 1, • x3 − x2 − 3x + 1. The first 30 complex cubic fields (ordered by discriminant) have class number one. These are the cubic fields of discriminant between 0 and −268 (inclusively). The next complex cubic field (of discriminant −283) has class number two. The negative discriminants less than 150 with class number one are: • −23, −31, −44, −59, −76, −83, −87, −104, −107, −108, −116, −135, −139, −140. Polynomials defining the first three are respectively: • x3 − x2 + 1, • x3 + x − 1, • x3 − x2 + x + 1.[5]
136.4
Cyclotomic fields
The following is a complete list of n for which the field Q(ζn) has class number 1:[6] • 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84.[7] On the other hand, the maximal real subfields Q(cos(2π/2n )) of the 2-power cyclotomic fields Q(ζ₂n ) (where n is a positive integer) are known to have class number 1 for n≤8,[8] and it is conjectured that they have class number 1 for all n. Weber showed that these fields have odd class number. In 2009, Fukuda and Komatsu showed that the class numbers of these fields have no prime factor less than 107 ,[9] and later improved this bound to 109 .[10] These fields are the n-th layers of the cyclotomic Z2 -extension of Q. Also in 2009, Morisawa showed that the class numbers of the layers of the cyclotomic Z3 -extension of Q have no prime factor less than 104 .[11] Coates has raised the question of whether, for all primes p, every layer of the cyclotomic Zp-extension of Q has class number 1.
136.5
CM fields
Simultaneously generalizing the case of imaginary quadratic fields and cyclotomic fields is the case of a CM field K, i.e. a totally imaginary quadratic extension of a totally real field. In 1974, Harold Stark conjectured that there are finitely many CM fields of class number 1.[12] He showed that there are finitely many of a fixed degree. Shortly thereafter, Andrew Odlyzko showed that there are only finitely many Galois CM fields of class number 1.[13] In 2001, V. Kumar Murty showed that of all CM fields whose Galois closure has solvable Galois group, only finitely many have class number 1.[14] A complete list of the 172 abelian CM fields of class number 1 was determined in the early 1990s by Ken Yamamura and is available on pages 915–919 of his article on the subject.[15] Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of class number 1.[16]
136.6. SEE ALSO
431
136.6 See also • Class number problem • Class number formula
136.7 Notes [1] Chapter I, section 6, p. 37 of Neukirch 1999 √ [2] Dembélé, Lassina (2005). “Explicit computations of Hilbert modular forms on Q( 5) " (PDF). Exp. Math. 14 (4): 457–466. doi:10.1080/10586458.2005.10128939. ISSN 1058-6458. Zbl 1152.11328. [3] H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer Verlag (1993), Appendix B2, p.507 [4] H. Cohen and H. W. Lenstra, Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, Proc. 13th Journées Arithmétiques, ed. H. Jager, Lect. Notes in Math. 1068, Springer-Verlag, 1984, pp. 33—62 [5] Tables available at [6] Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Graduate Texts in Mathematics 83 (2nd ed.). SpringerVerlag. Theorem 11.1. ISBN 0-387-94762-0. Zbl 0966.11047. [7] Note that values of n congruent to 2 modulo 4 are redundant since Q(ζ2n) = Q(ζn) when n is odd. [8] J. C. Miller, Class numbers of totally real fields and applications to the Weber class number problem, http://arxiv.org/abs/ 1405.1094 [9] Fukuda, Takashi; Komatsu, Keiichi (2009). “Weber’s class number problem in the cyclotomic Z2 -extension of Q ". Exp. Math. 18 (2): 213–222. ISSN 1058-6458. MR 2549691. Zbl 1189.11033. [10] Fukuda, Takashi; Komatsu, Keiichi (2011). “Weber’s class number problem in the cyclotomic Z2 -extension of Q III”. Int. J. Number Theory 7 (6): 1627–1635. doi:10.1142/S1793042111004782. ISSN 1793-7310. MR 2835816. Zbl 1226.11119. [11] Morisawa, Takayuki (2009). “A class number problem in the cyclotomic Z3 -extension of Q ". Tokyo J. Math. 32 (2): 549–558. doi:10.3836/tjm/1264170249. ISSN 0387-3870. MR 2589962. Zbl 1205.11116. [12] Stark, Harold (1974), “Some effective cases of the Brauer–Siegel theorem”, Inventiones Mathematicae 23: 135–152, doi:10.1007/bf01405166 [13] Odlyzko, Andrew (1975), “Some analytic estimates of class numbers and discriminants”, Inventiones Mathematicae 29 (3): 275–286, doi:10.1007/bf01389854 [14] Murty, V. Kumar (2001), “Class numbers of CM-fields with solvable normal closure”, Compositio Mathematica 127 (3): 273–287 [15] Yamamura, Ken (1994), “The determination of the imaginary abelian number fields with class number one”, Mathematics of Computation 62 (206): 899–921, doi:10.2307/2153549 [16] Louboutin, Stéphane; Okazaki, Ryotaro (1994), “Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one”, Acta Arithmetica 67 (1): 47–62
136.8 References • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
Chapter 137
Local class field theory In mathematics, local class field theory, introduced by Helmut Hasse,[1] is the study of abelian extensions of local fields; here, “local field” means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Qp (where p is any prime number), or a finite extension of the field of formal Laurent series Fq((T)) over a finite field Fq.[2] It is the analogue for local fields of global class field theory.
137.1 Connection to Galois groups Local class field theory gives a description of the Galois group G of the maximal abelian extension of a local field K via the reciprocity map which acts from the multiplicative group K × =K\{0}. For a finite abelian extension L of K the reciprocity map induces an isomorphism of the quotient group K × /N(L× ) of K × by the norm group N(L× ) of the extension L× to the Galois group Gal(L/K) of the extension.[3] The absolute Galois group G of K is compact and the group K × is not compact. Taking the case where K is a finite extension of the p-adic numbers Q or formal power series over a finite field, the group K × is the product of a compact group with an infinite cyclic group Z. The main topological operation is to replace K × by its profinite completion, which is roughly the same as replacing the factor Z by its profinite completion Z^ . The profinite completion of K × is the group isomorphic with G via the local reciprocity map. The actual isomorphism used and the existence theorem is described in the theory of the norm residue symbol. There are several different approaches to the theory, using central division algebras or Tate cohomology or an explicit description of the reciprocity map. There are also two different normalizations of the reciprocity map: in the case of an unramified extension, one of them asks that the (arithmetic) Frobenius element corresponds to the elements of “K” of valuation 1; the other one is the opposite.
137.2 Lubin–Tate theory Main article: Lubin–Tate theory Lubin–Tate theory is important in explicit local class field theory. The unramified part of any abelian extension is easily constructed, Lubin–Tate finds its value in producing the ramified part. This works by defining a family of modules (indexed by the natural numbers) over the ring of integers consisting of what can be considered as roots of the power series repeatedly composed with itself. The compositum of all fields formed by adjoining such modules to the original field gives the ramified part. A Lubin–Tate extension of a local field K is an abelian extension of K obtained by considering the p-division points of a Lubin–Tate group. If g is an Eisenstein polynomial, f(t) = t g(t) and F the Lubin–Tate formal group, let θn denote a root of gf n−1 (t)=g(f(f(⋯(f(t))⋯))). Then K(θn) is an abelian extension of K with Galois group isomorphic to U/1+pn where U is the unit group of the ring of integers of K and p is the maximal ideal.[4] 432
137.3. HIGHER LOCAL CLASS FIELD THEORY
433
137.3 Higher local class field theory For a higher-dimensional local field K there is a higher local reciprocity map which describes abelian extensions of the field in terms of open subgroups of finite index in the Milnor K-group of the field. Namely, if K is an n -dimensional local field then one uses KM n (K) or its separated quotient endowed with a suitable topology. When n = 1 the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if n > 1 . Higher-dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and K. Kato, I. Fesenko, Sh. Saito in the general case.
137.4 See also • Quasi-finite field • Local Langlands conjectures • Norm group
137.5 References [1] Hasse, H. (1930), “Die Normenresttheorie relativ-Abelscher Zahlkörper als Klassenkörpertheorie im Kleinen.”, Journal für die reine und angewandte Mathematik (in German) 162: 145–154, doi:10.1515/crll.1930.162.145, ISSN 0075-4102, JFM 56.0165.03 [2] Kostrikin, A.I.; Shafarevich, I.R. (1996), Algebra IX: Finite Groups of Lie Type Finite-Dimensional Division Algebras, Encyclopedia of Mathematics 77, Springer Science & Business Media [3] Fesenko, Ivan and Vostokov, Sergei, Local Fields and their Extensions, 2nd ed., American Mathematical Society, 2002, ISBN 0-8218-3259-X [4] Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. pp. 62–63. ISBN 3-540-63003-1. Zbl 0819.11044.
137.6 Further reading • Fesenko, Ivan B.; Kurihara, Masato, eds. (2000), Invitation to Higher Local Fields, Geometry and Topology Monographs 3 (First ed.), University of Warwick: Mathematical Sciences Publishers, doi:10.2140/gtm.2000.3, ISSN 1464-8989, Zbl 0954.00026 • Iwasawa, Kenkichi (1986), Local class field theory, Oxford Science Publications, The Clarendon Press Oxford University Press, ISBN 978-0-19-504030-2, MR 863740 • Milne, James, Class Field Theory. • Neukirch, Jürgen (1986), Class field theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 280, Berlin, New York: Springer-Verlag, ISBN 978-3-540-15251-4, MR 819231 • Serre, Jean-Pierre (1967), “Local class field theory”, in Cassels, John William Scott; Fröhlich, Albrecht, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 128– 161, ISBN 978-0-9502734-2-6, MR 0220701 • Serre, Jean-Pierre (1979) [1962], Corps Locaux (English translation: Local Fields), Graduate Texts in Mathematics 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 0150130
Chapter 138
Local Euler characteristic formula In the mathematical field of Galois cohomology, the local Euler characteristic formula is a result due to John Tate that computes the Euler characteristic of the group cohomology of the absolute Galois group GK of a nonarchimedean local field K.
138.1 Statement Let K be a non-archimedean local field, let Ks denote a separable closure of K, let GK = Gal(Ks /K) be the absolute Galois group of K, and let Hi (K, M) denote the group cohomology of GK with coefficients in M. Since the cohomological dimension of GK is two,[1] Hi (K, M) = 0 for i ≥ 3. Therefore, the Euler characteristic only involves the groups with i = 0, 1, 2.
138.1.1
Case of finite modules
Let M be a GK-module of finite order m. The Euler characteristic of M is defined to be[2]
χ(GK , M ) =
#H 0 (K, M ) · #H 2 (K, M ) #H 1 (K, M )
(the ith cohomology groups for i ≥ 3 appear tacitly as their sizes are all one). Let R denote the ring of integers of K. Tate’s result then states that if m is relatively prime to the characteristic of K, then[3] −1
χ(GK , M ) = (#R/mR)
,
i.e. the inverse of the order of the quotient ring R/mR. Two special cases worth singling out are the following. If the order of M is relatively prime to the characteristic of the residue field of K, then the Euler characteristic is one. If K is a finite extension of the p-adic numbers Qp, and if vp denotes the p-adic valuation, then χ(GK , M ) = p−[K:Qp ]vp (m) where [K:Qp] is the degree of K over Qp. The Euler characteristic can be rewritten, using local Tate duality, as
χ(GK , M ) =
#H 0 (K, M ) · #H 0 (K, M ′ ) #H 1 (K, M )
where M ′ is the local Tate dual of M. 434
138.2. NOTES
435
138.2 Notes [1] Serre 2002, §II.4.3 [2] The Euler characteristic in a cohomology theory is normally written as an alternating sum of the sizes of the cohomology groups. In this case, the alternating product is more standard. [3] Milne 2006, Theorem I.2.8
138.3 References • Milne, James S. (2006), Arithmetic duality theorems (second ed.), Charleston, SC: BookSurge, LLC, ISBN 1-4196-4274-X, MR 2261462, retrieved 2010-03-27 • Serre, Jean-Pierre (2002), Galois cohomology, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42192-4, MR 1867431, translation of Cohomologie Galoisienne, SpringerVerlag Lecture Notes 5 (1964).
Chapter 139
Local field In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.[1] Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and those in which it is not. In the first case, one calls the local field an archimedean local field, in the second case, one calls it a non-archimedean local field. Local fields arise naturally in number theory as completions of global fields. Every local field is isomorphic (as a topological field) to one of the following: • Archimedean local fields (characteristic zero): the real numbers R, and the complex numbers C. • Non-archimedean local fields of characteristic zero: finite extensions of the p-adic numbers Qp (where p is any prime number). • Non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). There is an equivalent definition of non-archimedean local field: it is a field that is complete with respect to a discrete valuation and whose residue field is finite. However, some authors consider a more general notion, requiring only that the residue field be perfect, not necessarily finite.[2] This article uses the former definition.
139.1 Induced absolute value Given a locally compact topological field K, an absolute value can be defined as follows. First, consider the additive group of the field. As a locally compact topological group, it has a unique (up to positive scalar multiple) Haar measure μ. The absolute value is defined so as to measure the change in size of a set after multiplying it by an element of K. Specifically, define |·| : K → R by[3]
|a| :=
µ(aX) µ(X)
for any measurable subset X of K (with 0 < μ(X) < ∞). This absolute value does not depend on X nor on the choice of Haar measure (since the same scalar multiple ambiguity will occur in both the numerator and the denominator). This definition is very similar to that of the modular function. Given such an absolute value on K, a new induced topology can be defined on K. This topology is the same as the original topology.[4] Explicitly, for a positive real number m, define the subset B of K by
Bm := {a ∈ K : |a| ≤ m}. Then, the B make up a neighbourhood basis of 0 in K. 436
139.2. NON-ARCHIMEDEAN LOCAL FIELD THEORY
437
139.2 Non-archimedean local field theory For a non-archimedean local field F (with absolute value denoted by |·|), the following objects are important: • its ring of integers O = {a ∈ F : |a| ≤ 1} which is a discrete valuation ring, is the closed unit ball of F, and is compact; • the units in its ring of integers O× = {a ∈ F : |a| = 1} which forms a group and is the unit sphere of F; • the unique non-zero prime ideal m in its ring of integers which is its open unit ball {a ∈ F : |a| < 1} ; • a generator ϖ of m called a uniformizer of F; • its residue field k = O/m which is finite (since it is compact and discrete). Every non-zero element a of F can be written as a = ϖn u with u a unit, and n a unique integer. The normalized valuation of F is the surjective function v : F → Z ∪ {∞} defined by sending a non-zero a to the unique integer n such that a = ϖn u with u a unit, and by sending 0 to ∞. If q is the cardinality of the residue field, the absolute value on F induced by its structure as a local field is given by[5] |a| = q −v(a) . An equivalent definition of a non-archimedean local field is that it is a field that is complete with respect to a discrete valuation and whose residue field is finite.
139.2.1
Examples
1. The p-adic numbers: the ring of integers of Qp is the ring of p-adic integers Zp. Its prime ideal is pZp and its residue field is Z/pZ. Every non-zero element of Q can be written as u pn where u is a unit in Zp and n is an integer, then v(u pn ) = n for the normalized valuation. 2. The formal Laurent series over a finite field: the ring of integers of Fq((T)) is the ring of formal power series Fq[[T]]. Its prime ideal is (T) (i.e. the power series whose constant term is zero) and its residue field is Fq. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows: (∑∞ ) i v = −m (where a₋m is non-zero). i=−m ai T 3. The formal Laurent series over the complex numbers is not a local field. For example, its residue field is C[[T]]/(T) = C, which is not finite.
139.2.2
Higher unit groups
The nth higher unit group of a non-archimedean local field F is { } U (n) = 1 + mn = u ∈ O× : u ≡ 1 (mod mn ) for n ≥ 1. The group U (1) is called the group of principal units, and any element of it is called a principal unit. The full unit group O× is denoted U (0) . The higher unit groups provide a decreasing filtration of the unit group O× ⊇ U (1) ⊇ U (2) ⊇ · · · whose quotients are given by × O× /U (n) ∼ = (O/mn ) and U (n) /U (n+1) ≈ O/m
for n ≥ 1.[6] (Here " ≈ " means a non-canonical isomorphism.)
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CHAPTER 139. LOCAL FIELD
139.2.3
Structure of the unit group
The multiplicative group of non-zero elements of a non-archimedean local field F is isomorphic to F× ∼ = (ϖ) × µq−1 × U (1) where q is the order of the residue field, and μq₋₁ is the group of (q−1)st roots of unity (in F). Its structure as an abelian group depends on its characteristic: • If F has positive characteristic p, then
F× ∼ = Z ⊕ Z/(q − 1) ⊕ ZN p where N denotes the natural numbers; • If F has characteristic zero (i.e. it is a finite extension of Qp of degree d), then
F× ∼ = Z ⊕ Z/(q − 1) ⊕ Z/pa ⊕ Zdp where a ≥ 0 is defined so that the group of p-power roots of unity in F is µpa .[7]
139.3 Higher-dimensional local fields Main article: Higher local field It is natural to introduce non-archimedean local fields in a uniform geometric way as the field of fractions of the completion of the local ring of a one-dimensional arithmetic scheme of rank 1 at its non-singular point. For generalizations, a local field is sometimes called a one-dimensional local field. For a non-negative integer n, an n-dimensional local field is a complete discrete valuation field whose residue field is an (n − 1)-dimensional local field.[8] Depending on the definition of local field, a zero-dimensional local field is then either a finite field (with the definition used in this article), or a quasi-finite field,[9] or a perfect field. From the geometric point of view, n-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an n-dimensional arithmetic scheme.
139.4 See also • Hasse principle • Local class field theory
139.5 Notes [1] Page 20 of Weil 1995 [2] See, for example, definition 1.4.6 of Fesenko & Vostokov 2002 [3] Page 4 of Weil 1995 [4] Corollary 1, page 5 of Weil 1995
139.6. REFERENCES
439
[5] Weil 1995, chapter I, theorem 6 [6] Neukirch 1999, p. 122 [7] Neukirch 1999, theorem II.5.7 [8] Definition 1.4.6 of Fesenko & Vostokov 2002 [9] Serre 1995
139.6 References • Serre, Jean-Pierre (1995), Local Fields, Graduate texts in mathematics 67, Berlin, Heidelberg: SpringerVerlag, ISBN 0-387-90424-7 • Weil, André (1995), Basic number theory, Classics in Mathematics, Berlin, Heidelberg: Springer-Verlag, ISBN 3-540-58655-5 • Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3259-2, MR 1915966 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
139.7 Further reading • A. Frohlich, “Local fields”, in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.I • Milne, James, Algebraic Number Theory. • Schikhoff, W.H. (1984) Ultrametric Calculus
139.8 External links • Hazewinkel, Michiel, ed. (2001), “Local field”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4
Chapter 140
Local Fields Corps Locaux by Jean-Pierre Serre, originally published in 1962 and translated into English as Local Fields by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text covering local fields, ramification, group cohomology, and local class field theory. The book’s end goal is to present local class field theory from the cohomological point of view. This theory concerns extensions of “local” (i.e., complete for a discrete valuation) fields with finite residue field.
140.1 Contents 1. Part I, Local Fields (Basic Facts): Discrete valuation rings, Dedekind domains, and Completion. 2. Part II, Ramification: Discriminant & Different, Ramification Groups, The Norm, and Artin Representation. 3. Part III, Group Cohomology: Abelian & Nonabelian Cohomology, Cohomology of Finite Groups, Theorems of Tate and Nakayama, Galois Cohomology, Class Formations, and Computation of Cup Products. 4. Part IV, Local Class Field Theory: Brauer Group of a Local Field, Local Class Field Theory, Local Symbols and Existence Theorem, and Ramification.
140.2 References • Serre, Jean-Pierre (1980), Local Fields, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 0554237
440
Chapter 141
Local Langlands conjectures In mathematics, the local Langlands conjectures, introduced by Langlands (1967, 1970), are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.
141.1 Local Langlands conjectures for GL1 The local Langlands conjectures for GL1 (K) follow from (and are essentially equivalent to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL1 (K)= K * to the abelianization of the Weil group. In particular irreducible smooth representations of GL1 (K) are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL1 (C). This gives the Langlands correspondence between homomorphisms of the Weil group to GL1 (C) and irreducible smooth representations of GL1 (K).
141.2 Representations of the Weil group Representations of the Weil group do not quite correspond to irreducible smooth representations of general linear groups. To get a bijection, one has to slightly modify the notion of a representation of the Weil group, to something called a Weil–Deligne representation. This consists of a representation of the Weil group on a vector space V together with a nilpotent endomorphism N of V such that wNw−1 =||w||N, or equivalently a representation of the Weil–Deligne group. In addition the representation of the Weil group should have an open kernel, and should be (Frobenius) semisimple. For every Frobenius semisimple complex n-dimensional Weil–Deligne representations ρ of the Weil group of F there is an L-function L(s,ρ) and a local ε-factor ε(s,ρ,ψ) (depending on a character ψ of F).
141.3 Representations of GLn(F) The representations of GLn(F) appearing in the local Langlands correspondence are smooth irreducible complex representations. • “Smooth” means that every vector is fixed by some open subgroup. • “Irreducible” means that the representation is nonzero and has no subrepresentations other than 0 and itself. Smooth irreducible representations are automatically admissible. The Bernstein–Zelevinsky classification reduces the classification of irreducible smooth representations to cuspidal representations. 441
442
CHAPTER 141. LOCAL LANGLANDS CONJECTURES
For every irreducible admissible complex representation π there is an L-function L(s,π) and a local ε-factor ε(s,π,ψ) (depending on a character ψ of F). More generally, if there are two irreducible admissible representations π and π' of general linear groups there are local Rankin–Selberg convolution L-functions L(s,π×π') and ε-factors ε(s,π×π',ψ). Bushnell & Kutzko (1993) described the irreducible admissible representations of general linear groups over local fields.
141.4 Local Langlands conjectures for GL2 The local Langlands conjecture for GL2 of a local field says that there is a (unique) bijection π from 2-dimensional semisimple Deligne representations of the Weil group to irreducible smooth representations of GL2 (F) that preserves L-functions, ε-factors, and commutes with twisting by characters of F * . Jacquet & Langlands (1970) verified the local Langlands conjectures for GL2 in the case when the residue field does not have characteristic 2. In this case the representations of the Weil group are all of cyclic or dihedral type. Gelfand & Graev (1962) classified the smooth irreducible representations of GL2 (F) when F has odd residue characteristic (see also (Gelfand, Graev & Pyatetskii-Shapiro 1969, chapter 2)), and claimed incorrectly that the classification for even residue characteristic differs only insignifictanly from the odd residue characteristic case. Weil (1974) pointed out that when the residue field has characteristic 2, there are some extra exceptional 2-dimensional representations of the Weil group whose image in PGL2 (C) is of tetrahedral or octahedral type. (For global Langlands conjectures, 2-dimensional representations can also be of icosahedral type, but this cannot happen in the local case as the Galois groups are solvable.) Tunnell (1978) proved the local Langlands conjectures for the general linear group GL2 (K) over the 2-adic numbers, and over local fields containing a cube root of unity. Kutzko (1980, 1980b) proved the local Langlands conjectures for the general linear group GL2 (K) over all local fields. Cartier (1981) and Bushnell & Henniart (2006) gave expositions of the proof.
141.5 Local Langlands conjectures for GL The local Langlands conjectures for general linear groups state that there are unique bijections π ↔ ρπ from equivalence classes of irreducible admissible representations π of GLn(F) to equivalence classes of continuous Frobenius semisimple complex n-dimensional Weil–Deligne representations ρπ of the Weil group of F, that preserve L-functions and ε-factors of pairs of representations, and coincide with the Artin map for 1-dimensional representations. In other words, • L(s,ρπ⊗ρπ') = L(s,π×π') • ε(s,ρπ⊗ρπ',ψ) = ε(s,π×π',ψ) Laumon, Rapoport & Stuhler (1993) proved the local Langlands conjectures for the general linear group GLn(K) for positive characteristic local fields K. Carayol (1992) gave an exposition of their work. Richard Taylor and Michael Harris (2001) proved the local Langlands conjectures for the general linear group GLn(K) for characteristic 0 local fields K. Henniart (2001) gave another proof. Carayol (2000) and Wedhorn (2008) gave expositions of their work.
141.6 Local Langlands conjectures for other groups Borel (1979) and Vogan (1993) discuss the Langlands conjectures for more general groups. The Langlands conjectures for arbitrary reductive groups G are more complicated to state than the ones for general linear groups, and it is unclear what the best way of stating them should be. Roughly speaking, admissible representations of a reductive group are grouped into disjoint finite sets called L-packets, which should correspond to some classes of homomorphisms, called L-parameters, from the local Langlands group to the L-group of G. Some earlier versions used the Weil−Deligne group or the Weil group instead of the local Langlands group, which gives a slightly weaker form of the conjecture.
141.7. REFERENCES
443
Langlands (1989) proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible admissible representations (up to infinitesimal equivalence), or, equivalently, of their irreducible (g, K) -modules. Gan & Takeda (2011) proved the local Langlands conjectures for the symplectic similitude group GSp(4) and used that in Gan & Takeda (2010) to deduce it for the symplectic group Sp(4).
141.7 References • Borel, Armand (1979), “Automorphic L-functions”, in Borel, Armand; Casselman, W., Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, ISBN 978-0-8218-1437-6, MR 546608 • Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 335, Berlin, New York: SpringerVerlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120 • Bushnell, Colin J.; Kutzko, Philip C. (1993), The admissible dual of GL(N) via compact open subgroups, Annals of Mathematics Studies 129, Princeton University Press, ISBN 978-0-691-03256-6, MR 1204652 • Carayol, Henri (1992), “Variétés de Drinfeld compactes, d'après Laumon, Rapoport et Stuhler”, Astérisque 206: 369–409, ISSN 0303-1179, MR 1206074 • Carayol, Henri (2000), “Séminaire Bourbaki. Vol. 1998/99.”, Astérisque 266: 191–243, ISSN 0303-1179, MR 1772675 |chapter= ignored (help) • Cartier, Pierre (1981), “La conjecture locale de Langlands pour GL(2) et la démonstration de Ph. Kutzko”, Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math. (in French) 842, Berlin, New York: Springer-Verlag, pp. 112–138, doi:10.1007/BFb0089931, ISBN 978-3-540-10292-2, MR 636520 • Gan, Wee Teck; Takeda, Shuichiro (2010), “The local Langlands conjecture for Sp(4)", International Mathematics Research Notices 2010 (15): 2987–3038, doi:10.1093/imrn/rnp203, ISSN 1073-7928, MR 2673717[[arXiv]]:[[arXiv: 0805.2731|0805.2731]] • Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) [1966], Representation theory and automorphic functions, Generalized functions 6, Philadelphia, Pa.: W. B. Saunders Co., ISBN 978-0-12-279506-0, MR 0220673 • Gan, Wee Teck; Takeda, Shuichiro (2011), “The local Langlands conjecture for GSp(4)", Annals of Mathematics 173 (3): 1841–1882, arXiv:0706.0952v1, doi:10.4007/annals.2011.173.3.12 • Harris, Michael; Taylor, Richard (2001), The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies 151, Princeton University Press, ISBN 978-0-691-09090-0, MR 1876802 • Henniart, Guy (2000), “Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique”, Inventiones Mathematicae 139 (2): 439–455, doi:10.1007/s002220050012, ISSN 0020-9910, MR 1738446 • Henniart, Guy (2006), “On the local Langlands and Jacquet-Langlands correspondences”, in Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis et al., International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 1171–1182, ISBN 978-3-03719-022-7, MR 2275640 • Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, MR 0401654 • Kudla, Stephen S. (1994), “The local Langlands correspondence: the non-Archimedean case”, in Jannsen, Uwe; Kleiman, Steven; Serre, Jean-Pierre, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Providence, R.I.: American Mathematical Society, pp. 365–391, ISBN 978-0-8218-1637-0, MR 1265559 • Kutzko, Philip (1980), “The Langlands conjecture for GL2 of a local field”, American Mathematical Society. Bulletin. New Series 2 (3): 455–458, doi:10.1090/S0273-0979-1980-14765-5, ISSN 0002-9904, MR 561532
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CHAPTER 141. LOCAL LANGLANDS CONJECTURES
• Kutzko, Philip (1980), “The Langlands conjecture for Gl2 of a local field”, Annals of Mathematics. Second Series 112 (2): 381–412, doi:10.2307/1971151, ISSN 0003-486X, MR 592296 • Langlands, Robert (1967), Letter to Prof. Weil • Langlands, R. P. (1970), “Problems in the theory of automorphic forms”, Lectures in modern analysis and applications, III, Lecture Notes in Math 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614 • Langlands, Robert P. (1989) [1973], “On the classification of irreducible representations of real algebraic groups”, in Sally, Paul J.; Vogan, David A., Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr. 31, Providence, R.I.: American Mathematical Society, pp. 101–170, ISBN 978-0-8218-1526-7, MR 1011897 • Laumon, G.; Rapoport, M.; Stuhler, U. (1993), “D-elliptic sheaves and the Langlands correspondence”, Inventiones Mathematicae 113 (2): 217–338, doi:10.1007/BF01244308, ISSN 0020-9910, MR 1228127 • Tunnell, Jerrold B. (1978), “On the local Langlands conjecture for GL(2)", Inventiones Mathematicae 46 (2): 179–200, doi:10.1007/BF01393255, ISSN 0020-9910, MR 0476703 • Vogan, David A. (1993), “The local Langlands conjecture”, in Adams, Jeffrey; Herb, Rebecca; Kudla, Stephen; Li, Jian-Shu; Lipsman, Ron; Rosenberg, Jonathan, Representation theory of groups and algebras, Contemp. Math. 145, Providence, R.I.: American Mathematical Society, pp. 305–379, ISBN 978-0-8218-5168-5, MR 1216197 • Wedhorn, Torsten (2008), “The local Langlands correspondence for GL(n) over p-adic fields”, in Göttsche, Lothar; Harder, G.; Raghunathan, M. S., School on Automorphic Forms on GL(n), ICTP Lect. Notes 21, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 237–320, arXiv:math/0011210, ISBN 978-92-95003-37-8, MR 2508771 • Weil, André (1974), “Exercices dyadiques”, Inventiones Mathematicae 27 (1-2): 1–22, doi:10.1007/BF01389962, ISSN 0020-9910, MR 0379445
141.8 External links • Harris, Michael (2000), The local Langlands correspondence, Notes of (half) a course at the IHP • The work of Robert Langlands • Automorphic Forms - The local Langlands conjecture Lecture by Richard Taylor
Chapter 142
Lubin–Tate formal group law In mathematics, the Lubin–Tate formal group law is a formal group law introduced by Lubin and Tate (1965) to isolate the local field part of the classical theory of complex multiplication of elliptic functions. In particular it can be used to construct the totally ramified abelian extensions of a local field. It does this by considering the (formal) endomorphisms of the formal group, emulating the way in which elliptic curves with extra endomorphisms are used to give abelian extensions of global fields
142.1 Definition of formal groups Let Zp be the ring of p-adic integers. The Lubin–Tate formal group law is the unique (1-dimensional) formal group law F such that e(x) = px + xp is an endomorphism of F, in other words
e(F (x, y)) = F (e(x), e(y)). More generally, the choice for e may be any power series such that e(x) = px + higher-degree terms and e(x) = xp mod p. All such group laws, for different choices of e satisfying these conditions, are strictly isomorphic.[1] We choose these conditions so as to ensure that they reduce modulo the maximal ideal to Frobenius and the derivative at the origin is the prime element. For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax + higher-degree terms. This gives an action of the ring Zp on the Lubin–Tate formal group law. There is a similar construction with Zp replaced by any complete discrete valuation ring with finite residue class field, where p is replaced by a choice of uniformizer.[2]
142.2 Example We outline here a formal group equivalent of the Frobenius element, which is of great importance in class field theory,[3] generating the maximal unramified extension as the image of the reciprocity map. For this example we need the notion of an endomorphism of formal groups, which is a formal group homomorphism f where the domain is the codomain. A formal group homomorphism from a formal group F to a formal group G is a power series over the same ring as the formal groups which has zero constant term and is such that:
f (F (X, Y )) = G(f (X), f (Y )) 445
446
CHAPTER 142. LUBIN–TATE FORMAL GROUP LAW
Consider a formal group F(X,Y) with coefficients in the ring of integers in a local field (for example Zp), taking X and Y to be in the unique maximal ideal gives us a convergent power series and in this case we define F(X,Y) = X +F Y and we have a genuine group law. For example if F(X,Y)=X+Y, then this is the usual addition. This is isomorphic to the case of F(X,Y)=X+Y+XY, where we have multiplication on the set of elements which can be written as 1 added to an element of the prime ideal. In the latter case f(S) = (I + S)p −1 is an endomorphism of F and the isomorphism identifies f with the Frobenius element.
142.3 Generating ramified extensions Lubin–Tate theory is important in explicit local class field theory. The unramified part of any abelian extension is easily constructed, Lubin–Tate finds its value in producing the ramified part. This works by defining a family of modules (indexed by the natural numbers) over the ring of integers consisting of what can be considered as roots of the power series repeatedly composed with itself. The compositum of all fields formed by adjoining such modules to the original field gives the ramified part. A Lubin–Tate extension of a local field K is an abelian extension of K obtained by considering the p-division points of a Lubin–Tate group. If g is an Eisenstein polynomial, f(t) = t g(t) and F the Lubin–Tate formal group, let θn denote a root of gf n−1 (t)=g(f(f(⋯(f(t))⋯))). Then K(θn) is an abelian extension of K with Galois group isomorphic to U/1+pn where U is the unit group of the ring of integers of K and p is the maximal ideal.[2]
142.4 Connection with stable homotopy theory Lubin and Tate studied the deformation theory of such formal groups. A later application of the theory has been in the field of stable homotopy theory, with the construction of a particular extraordinary cohomology theory associated to the construction for a given prime p. As part of general machinery for formal groups, a cohomology theory with spectrum is set up for the Lubin–Tate formal group, which also goes by the names of Morava E-theory or completed Johnson–Wilson theory.[4]
142.5 References Notes [1] Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences 49 (Second ed.). p. 168. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002. [2] Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 62 (2nd printing of 1st ed.). Springer-Verlag. pp. 62–63. ISBN 3-540-63003-1. Zbl 0819.11044. [3] e.g. Serre (1967). Hazewinkel, Michiel (1975). “Local class field theory is easy”. Advances in Math. 18 (2): 148–181. doi:10.1016/0001-8708(75)90156-5. Zbl 0312.12022. [4] http://www.math.harvard.edu/~{}lurie/252xnotes/Lecture22.pdf
Sources • de Shalit, Ehud (1987), Iwasawa theory of elliptic curves with complex multiplication. p-adic L functions, Perspectives in Mathematics 3, Academic Press, ISBN 0-12-210255-X, Zbl 0674.12004 • Iwasawa, Kenkichi (1986), Local class field theory, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-504030-2, MR 863740, Zbl 0604.12014 • Lubin, Jonathan; Tate, John (1965), “Formal complex multiplication in local fields”, Annals of Mathematics. Second Series 81: 380–387, ISSN 0003-486X, JSTOR 1970622, MR 0172878, Zbl 0128.26501 • Lubin, Jonathan; Tate, John (1966), “Formal moduli for one-parameter formal Lie groups”, Bulletin de la Société Mathématique de France 94: 49–59, ISSN 0037-9484, MR 0238854, Zbl 0156.04105
142.6. EXTERNAL LINKS
447
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Serre, Jean-Pierre (1967), “Local class field theory”, in Cassels, J.W.S.; Fröhlich, Albrecht, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Academic Press, pp. 128–161, MR 0220701, Zbl 0153.07403
142.6 External links • Lurie, J. (2010), Lubin–Tate theory (PDF)
Chapter 143
Lüroth’s theorem In mathematics, Lüroth’s theorem asserts that every field that lies between two other fields K and K(X) must be generated as an extension of K by a single element of K(X). This result is named after Jacob Lüroth, who proved it in 1876.[1]
143.1 Statement Let K be a field and M be an intermediate field between K and K(X) , for some indeterminate X. Then there exists a rational function f (X) ∈ K(X) such that M = K(f (X)) . In other words, every intermediate extension between K and K(X) is a simple extension.
143.2 Proofs The proof of Lüroth’s theorem can be derived easily from the theory of rational curves, using the geometric genus.[2] This method is non-elementary, but several short proofs using only the basics of field theory have long been known. Many of these simple proofs use Gauss’s lemma on primitive polynomials as a main step.[3]
143.3 References [1] Burau, Werner (2008), “Lueroth (or Lüroth), Jakob”, Complete Dictionary of Scientific Biography [2] Cohn, P. M. (1991), Algebraic Numbers and Algebraic Functions, Chapman Hall/CRC Mathematics Series 4, CRC Press, p. 148, ISBN 9780412361906. [3] E.g. see Mines, Ray; Richman, Fred (1988), A Course in Constructive Algebra, Universitext, Springer, p. 148, ISBN 9780387966403.
448
Chapter 144
Maillet’s determinant In mathematics, Maillet’s determinant Dp is the determinant of the matrix introduced by Maillet (1913) whose entries are R(s/r) for s,r = 1, 2, ..., (p – 1)/2 ∈ Z/pZ for an odd prime p, where and R(a) is the least positive residue of a mod p (Muir 1930, pages 340–342). Malo (1914) calculated the determinant Dp for p = 3, 5, 7, 11, 13 and found that in these cases it is given by (–p)(p – 3)/2 , and conjectured that it is given by this formula in general. Carlitz & Olson (1955) showed that this conjecture is incorrect; the determinant in general is given by Dp = (–p)(p – 3)/2 h− , where h− is the first factor of the class number of the cyclotomic field generated by pth roots of 1, which happens to be 1 for p less than 23. In particular this verifies Maillet’s conjecture that the determinant is always non-zero. Chowla and Weil had previously found the same formula but did not publish it.
144.1 References • Carlitz, L.; Olson, F. R. (1955), “Maillet’s determinant”, Proceedings of the American Mathematical Society 6: 265–269, doi:10.2307/2032352, ISSN 0002-9939, MR 0069207 • Maillet, E. (1913), “Question 4269”, L'Intermédiaire des Mathématiciens xx: 218 • Malo, E. (1914), L'Intermédiaire des Mathématiciens xxi: 173–176 Missing or empty |title= (help) • Muir, Thomas (1930), Contributions To The History Of Determinants 1900–1920, Blackie And Son Limited.
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Chapter 145
Minimal polynomial (field theory) For the minimal polynomial of a matrix, see Minimal polynomial (linear algebra). In field theory, a branch of mathematics, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E. The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let Jα be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in Jα. The set Jα is so named because it is an ideal of F[x]. The zero polynomial, whose every coefficient is 0, is in every Jα since 0αi = 0 for all α and i. This makes the zero polynomial useless for classifying different values of α into types, so it is excepted. If there are any non-zero polynomials in Jα, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in Jα. This is the minimal polynomial of α with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of Jα, then α is called a transcendental element over F and has no minimal polynomial with respect to E/F. Minimal polynomials are useful for constructing and analyzing field extensions. When α is algebraic with minimal polynomial a(x), the smallest field that contains both F and α is isomorphic to the quotient ring F[x]/⟨a(x)⟩, where ⟨a(x)⟩ is the ideal of F[x] generated by a(x). Minimal polynomials are also used to define conjugate elements.
145.1 Definition Let E/F be a field extension, α an element of E, and F[x] the ring of polynomials in x over F. The minimal polynomial of α is the monic polynomial of least degree among all polynomials in F[x] having α as a root; it exists when α is algebraic over F, that is, when f(α) = 0 for some non-zero polynomial f(x) in F[x].
145.1.1
Uniqueness
Let a(x) be the minimal polynomial of α with respect to E/F. The uniqueness of a(x) is established by considering the ring homomorphism subα from F[x] to E that substitutes α for x, that is, subα(f(x)) = f(α). The kernel of subα, ker(subα), is the set of all polynomials in F[x] that have α as a root. That is, ker(subα) = Jα from above. Since subα is a ring homomorphism, ker(subα) is an ideal of F[x]. Since F[x] is a principal ring whenever F is a field, there is at least one polynomial in ker(subα) that generates ker(subα). Such a polynomial will have least degree among all non-zero polynomials in ker(subα), and a(x) is taken to be the unique monic polynomial among these.
145.2 Properties A minimal polynomial is irreducible. Let E/F be a field extension over F as above, α ∈ E, and f ∈ F[x] a minimal polynomial for α. Suppose f = gh, where g,h ∈ F[x] are of lower degree than f. Now f(α) = 0. Since fields are also integral domains, we have g(α) = 0 or h(α) = 0. This contradicts the minimality of the degree of f. Thus minimal polynomials are irreducible. 450
145.3. EXAMPLES
451
145.3 Examples If F = Q, E = R, α = √2, then the minimal polynomial for α is a(x) = x2 − 2. The base field F is important as it determines the possibilities for the coefficients of a(x). For instance, if we take F = R, then the minimal polynomial for α = √2 is a(x) = x − √2. If α = √2 + √3, then the minimal polynomial in Q[x] is a(x) = x4 − 10x2 + 1 = (x − √2 − √3)(x + √2 − √3)(x − √2 + √3)(x + √2 + √3). The minimal polynomial in Q[x] of the sum of the square roots of the first n prime numbers is constructed analogously, and is called a Swinnerton-Dyer polynomial. The minimal polynomials in Q[x] of roots of unity are the cyclotomic polynomials.
145.4 References • Weisstein, Eric W., “Algebraic Number Minimal Polynomial”, MathWorld. • Minimal polynomial at PlanetMath.org. • Pinter, Charles C. A Book of Abstract Algebra. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270-273. ISBN 978-0-486-47417-5
Chapter 146
Minkowski space (number field) For Minkowski space-time, see Minkowski space. In mathematics, in the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field. If K is a number field of degree d then there are d distinct embeddings of K into C. We let KC be the image of K in the product Cd , considered as equipped with the usual Hermitian inner product. If c denotes complex conjugation, let KR denote the subspace of KC fixed by c, equipped with a scalar product. This is the Minkowski space of K.
146.1 References • Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften 322. Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
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Chapter 147
Mode of a linear field In physics, a vector field is linear if it is a solution of a set of linear equations E. For instance, in physics, the electromagnetic field in vacuum, defined in the usual (3 + 1)-dimensional space S, obeys Maxwell’s equations. A linear combination of electromagnetic fields, with constant, real coefficients, is a new field which obeys Maxwell’s equations. The solutions of the linear equations are represented in a real vector space M. A radius of M, which represents proportional solutions, is called a “mode”. A norm may be defined. For instance, in electromagnetism, it is usually the energy of the solution assuming that there is no other field in S. From the norm are defined the orthogonality and the scalar product of solutions. The orthogonality of solutions extends to the corresponding modes.
453
Chapter 148
Modulus (algebraic number theory) For the operation that gives a number’s remainder, see Modulo operation. In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle,[1] or extended ideal[2] ) is a formal product of places of a global field (i.e. an algebraic number field or a global function field). It is used to encode ramification data for abelian extensions of a global field.
148.1 Definition Let K be a global field with ring of integers R. A modulus is a formal product[3][4]
m=
∏
pν(p) , ν(p) ≥ 0
p
where p runs over all places of K, finite or infinite, the exponents ν(p) are zero except for finitely many p. If K is a number field, ν(p) = 0 or 1 for real places and ν(p) = 0 for complex places. If K is a function field, ν(p) = 0 for all infinite places. In the function field case, a modulus is the same thing as an effective divisor,[5] and in the number field case, a modulus can be considered as special form of Arakelov divisor.[6] The notion of congruence can be extended to the setting of moduli. If a and b are elements of K × , the definition of a ≡∗ b (mod pν ) depends on what type of prime p is:[7][8] • if it is finite, then
a ≡∗ b (mod pν ) ⇔ ordp
(a b
) −1 ≥ν
where ord is the normalized valuation associated to p; • if it is a real place (of a number field) and ν = 1, then
a ≡∗ b (mod p) ⇔
a >0 b
under the real embedding associated to p. • if it is any other infinite place, there is no condition. Then, given a modulus m, a ≡∗ b (mod m) if a ≡∗ b (mod pν(p) ) for all p such that ν(p) > 0. 454
148.2. RAY CLASS GROUP
455
148.2 Ray class group Main article: Ray class group The ray modulo m is[9][10][11] { } Km,1 = a ∈ K × : a ≡∗ 1 (mod m) . A modulus m can be split into two parts, m and m∞, the product over the finite and infinite places, respectively. Let I m to be one of the following: • if K is a number field, the subgroup of the group of fractional ideals generated by ideals coprime to m ;[12] • if K is a function field of an algebraic curve over k, the group of divisors, rational over k, with support away from m.[13] In both case, there is a group homomorphism i : K ,₁ → I m obtained by sending a to the principal ideal (resp. divisor) (a). The ray class group modulo m is the quotient C = I m / i(K ,₁).[14][15] A coset of i(K ,₁) is called a ray class modulo m. Erich Hecke's original definition of Hecke characters may be interpreted in terms of characters of the ray class group with respect to some modulus m.[16]
148.2.1
Properties
When K is a number field, the following properties hold.[17] • When m = 1, the ray class group is just the ideal class group. • The ray class group is finite. Its order is the ray class number. • The ray class number is divisible by the class number of K.
148.3 Notes [1] Lang 1994, §VI.1 [2] Cohn 1985, definition 7.2.1 [3] Janusz 1996, §IV.1 [4] Serre 1988, §III.1 [5] Serre 1988, §III.1 [6] Neukirch 1999, §III.1 [7] Janusz 1996, §IV.1 [8] Serre 1988, §III.1 [9] Milne 2008, §V.1 [10] Janusz 1996, §IV.1 [11] Serre 1988, §VI.6 [12] Janusz 1996, §IV.1
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[13] Serre 1988, §V.1 [14] Janusz 1996, §IV.1 [15] Serre 1988, §VI.6 [16] Neukirch 1999, §VII.6 [17] Janusz & 1996 §4.1
148.4 References • Cohn, Harvey (1985), Introduction to the construction of class fields, Cambridge studies in advanced mathematics 6, Cambridge University Press, ISBN 978-0-521-24762-7 • Janusz, Gerald J. (1996), Algebraic number fields, Graduate Studies in Mathematics 7, American Mathematical Society, ISBN 978-0-8218-0429-2 • Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics 110 (2 ed.), New York: SpringerVerlag, ISBN 978-0-387-94225-4, MR 1282723 • Milne, James (2008), Class field theory (v4.0 ed.), retrieved 2010-02-22 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics 117, New York: Springer-Verlag, ISBN 978-0-387-96648-9
Chapter 149
Monogenic field In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers OK is the polynomial ring Z[a]. The powers of such an element a constitute a power integral basis. In a monogenic field K, the field discriminant of K is equal to the discriminant of the minimal polynomial of α.
149.1 Examples Examples of monogenic fields include: • Quadratic fields: √ √ if K = Q( √ d) with d a square-free integer, then OK = Z[a] where a = (1 + d)/2 if d≡1 (mod 4) and a = d if d ≡ 2 or 3 (mod 4). • Cyclotomic fields: if K = Q(ζ) with ζ a root of unity, then OK = Z[ζ]. Also the maximal real subfield Q(ζ)+ = Q(ζ + ζ −1 ) is monogenic, with ring of integers Z[ζ + ζ −1 ]. While all quadratic fields are monogenic, already among cubic fields there are many that are not monogenic. The first example of a non-monogenic number field that was found is the cubic field generated by a root of the polynomial X 3 − X 2 − 2X − 8 , due to Richard Dedekind.
149.2 References • Narkiewicz, Władysław (2004). Elementary and Analytic Theory of Algebraic Numbers (3rd ed.). SpringerVerlag. p. 64. ISBN 3-540-21902-1. Zbl 1159.11039. • Gaál, István (2002). Diophantine Equations and Power Integral Bases. Boston, MA: Birkhäuser Verlag. ISBN 978-0-8176-4271-6. Zbl 1016.11059.
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Chapter 150
Multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: • the group under multiplication of the invertible elements of a field,[1] ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is (F ∖ {0}, •), where 0 refers to the zero element of F and the binary operation • is the field multiplication, • the algebraic torus GL(1).
150.1 Examples • The multiplicative group of integers modulo n is the group under multiplication of the invertible elements of Z/nZ . When n is not prime, there are elements other than zero that are not invertible. • The multiplicative group of positive real numbers, R+ , is an abelian group with 1 being its neutral element. The logarithm is a group isomorphism of this group to the additive group of real numbers, R .
150.2 Group scheme of roots of unity The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group GL(1), considered as a group scheme. That is, for any integer n > 1 we can consider the morphism on the multiplicative group that takes n-th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the morphism e that serves as the identity. The resulting group scheme is written μn. It gives rise to a reduced scheme, when we take it over a field K, if and only if the characteristic of K does not divide n. This makes it a source of some key examples of non-reduced schemes (schemes with nilpotent elements in their structure sheaves); for example μp over a finite field with p elements for any prime number p. This phenomenon is not easily expressed in the classical language of algebraic geometry. It turns out to be of major importance, for example, in expressing the duality theory of abelian varieties in characteristic p (theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing Kummer theory.
150.3 Notes [1] See Hazewinkel et al. (2004), p. 2.
150.4 References • Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, 458
150.5. SEE ALSO rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0
150.5 See also • Multiplicative group of integers modulo n • Additive group
459
Chapter 151
Nagata’s conjecture For the conjecture about curves, see Nagata’s conjecture on curves. In algebra, Nagata’s conjecture states that Nagata’s automorphism of the polynomial ring k[x,y,z] is wild. The conjecture was proposed by Nagata (1972) and proved by Ualbai U. Umirbaev and Ivan P. Shestakov (2004). Nagata’s automorphism is given by
(x, y, z) 7→ (x − 2(xz + y 2 )y − (xz + y 2 )2 z, y + (xz + y 2 )z, z).
151.1 References • Nagata, Masayoshi (1972), On automorphism group of k[x,y], Tokyo: Kinokuniya Book-Store Co. Ltd., MR 0337962 • Umirbaev, Ualbai U.; Shestakov, Ivan P. (2004), “The tame and the wild automorphisms of polynomial rings in three variables”, Journal of the American Mathematical Society 17 (1): 197–227, doi:10.1090/S0894-034703-00440-5, ISSN 0894-0347, MR 2015334
460
Chapter 152
Narrow class group In algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real numbers.
152.1 Formal definition Suppose that K is a finite extension of Q. Recall that the ordinary class group of K is defined to be
CK = IK /PK , where IK is the group of fractional ideals of K, and PK is the group of principal fractional ideals of K, that is, ideals of the form aOK where a is a unit of K. The narrow class group is defined to be the quotient
+ + CK = IK /PK ,
where now PK + is the group of totally positive principal fractional ideals of K; that is, ideals of the form aOK where a is a unit of K such that σ(a) is positive for every embedding
σ : K → R.
152.2 Uses The narrow class group features prominently in the theory of representing of integers by quadratic forms. An example is the following result (Fröhlich and Taylor, Chapter V, Theorem 1.25). Theorem. Suppose that √ K = Q( d), where d is a square-free integer, and that the narrow class group of K is trivial. Suppose that {ω1 , ω2 } is a basis for the ring of integers of K. Define a quadratic form qK (x, y) = NK/Q (ω1 x + ω2 y) 461
462
CHAPTER 152. NARROW CLASS GROUP where NK/Q is the norm. Then a prime number p is of the form p = qK (x, y) for some integers x and y if and only if either p | dK , or p=2
and dK ≡ 1 (mod 8),
or
(
p>2
and
dK p
) = 1,
where dK is the discriminant of K, and (a) b indicates the Legendre symbol.
152.2.1
Examples
For example, one can prove that the quadratic fields Q(√−1), Q(√2), Q(√−3) all have trivial narrow class group. Then, by choosing appropriate bases for the integers of each of these fields, the above theorem implies the following: • A prime p is of the form p = x2 + y2 for integers x and y if and only if
p=2
or p ≡ 1
(mod 4).
(This is known as Fermat’s theorem on sums of two squares.) • A prime p is of the form p = x2 − 2y2 for integers x and y if and only if
p=2
or p ≡ 1, 7
(mod 8).
• A prime p is of the form p = x2 − xy + y2 for integers x and y if and only if p = 3 or p ≡ 1 (mod 3). (cf. Eisenstein prime)
152.3 See also • Class group • Quadratic form
152.4 References • A. Fröhlich and M. J. Taylor, Algebraic Number Theory (p. 180), Cambridge University Press, 1991.
Chapter 153
Newton polygon
Construction of the Newton polygon of the polynomial P (X) = 1 + 5X + 15 X 2 + 35X 3 + 25X 5 + 625X 6 .
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring K[[X]], over K, where K was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms 463
464
CHAPTER 153. NEWTON POLYGON aXr
of the power series expansion solutions to equations P(F(X)) = 0 where P is a polynomial with coefficients in K[X], the polynomial ring; that is, implicitly defined algebraic functions. The exponents r here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in K[[Y]] with Y = X1/d for a denominator d corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating d. After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. Newton polygons have also been useful in the study of elliptic curves.
153.1 Definition A priori, given a polynomial over a field, the behaviour of the roots (assuming it has roots) will be unknown. Newton polygons provide one technique for the study of the behaviour of the roots. Let K be a local field with discrete valuation vK and let f (x) = an xn + · · · + a1 x + a0 ∈ K[x] with a0 an ̸= 0 . Then the Newton polygon of f is defined to be the lower convex hull of the set of points
Pi = (i, vK (ai )) , ignoring the points with ai = 0 . Restated geometrically, plot all of these points Pi on the xy-plane. Let’s assume that the points indices increase from left to right (P0 is the leftmost point, Pn is the rightmost point). Then, starting at P 0 , draw a ray straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point P 1 (not necessarily P 1 ). Break the ray here. Now draw a second ray from P 1 straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point P 2 . Continue until the process reaches the point Pn; the resulting polygon (containing the points P 0 , P 1 , P 2 , ..., P , Pn) is the Newton polygon. Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P 0 , ..., P . Stretch the band upwards, such that the band is stuck on its lower side by some of the points (the points act like nails, partially hammered into the xy plane). The vertices of the Newton polygon are exactly those points. For a neat diagram of this see Ch6 §3 of “Local Fields” by JWS Cassels, LMS Student Texts 3, CUP 1986. It is on p99 of the 1986 paperback edition.
153.2 History Newton polygons are named after Isaac Newton, who first described them and some of their uses in correspondence from the year 1676 addressed to Henry Oldenburg.[1]
153.3 Applications A Newton Polygon is sometimes a special case of a Newton Polytope, and can be used to construct asymptotic solutions of two-variable polynomial equations like 3x2 y 3 − xy 2 + 2x2 y 2 − x3 y = 0
153.3. APPLICATIONS
465
4
3
yn
xy
2
(3 x y - 1) x
2
y (3 y
2
- x)
2
-x y (y + x
2
)
1
0
0
1
x
2
m
3
4
This diagram shows the Newton polygon for P(x,y) = 3 x^2 y^3 - x y^2 + 2 x^2 y^2 - x^3 y, with positive monomials in red and negative monomials in cyan. Faces are labelled with the limiting terms they correspond to.
Another application of the Newton polygon comes from the following result: Let
µ1 , µ2 , . . . , µr be the slopes of the line segments of the Newton polygon of f (x) (as defined above) arranged in increasing order, and let
λ1 , λ2 , . . . , λr be the corresponding lengths of the line segments projected onto the x-axis (i.e. if we have a line segment stretching between the points Pi and Pj then the length is j − i ). Then for each integer 1 ≤ κ ≤ r , f (x) has exactly λκ roots with valuation −µκ .
466
CHAPTER 153. NEWTON POLYGON
153.4 Symmetric function explanation In the context of a valuation, we are given certain information in the form of the valuations of elementary symmetric functions of the roots of a polynomial, and require information on the valuations of the actual roots, in an algebraic closure. This has aspects both of ramification theory and singularity theory. The valid inferences possible are to the valuations first of the power sums, by means of Newton’s identities.
153.5 See also • Eisenstein’s criterion • Newton–Okounkov body
153.6 References [1] Egbert Brieskorn, Horst Knörrer (1986). Plane Algebraic Curves, pp. 370–383.
• Goss, David (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 35, Berlin, New York: Springer-Verlag, ISBN 978-3-540-61087-8, MR 1423131 • Gouvêa, Fernando: p-adic numbers: An introduction. Springer Verlag 1993. p. 199.
153.7 External links • Applet drawing a Newton Polygon
Chapter 154
Non-abelian class field theory In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field K, to the general Galois extension L/K. While class field theory was essentially known by 1930, the corresponding non-abelian theory has never been formulated in a definitive and accepted sense.[1]
154.1 History A presentation of class field theory in terms of group cohomology was carried out by Claude Chevalley, Emil Artin and others, mainly in the 1940s. This resulted in a formulation of the central results by means of the group cohomology of the idele class group. The theorems of the cohomological approach are independent of whether or not the Galois group G of L/K is abelian. This theory has never been regarded as the sought-after non-abelian theory. The first reason that can be cited for that is that it did not provide fresh information on the splitting of prime ideals in a Galois extension; a common way to explain the objective of a non-abelian class field theory is that it should provide a more explicit way to express such patterns of splitting.[2] The cohomological approach therefore was of limited use in even formulating non-abelian class field theory. Behind the history was the wish of Chevalley to write proofs for class field theory without using Dirichlet series: in other words to eliminate L-functions. The first wave of proofs of the central theorems of class field theory was structured as consisting of two 'inequalities’ (the same structure as in the proofs now given of the fundamental theorem of Galois theory, though much more complex). One of the two inequalities involved an argument with L-functions.[3] In a later reversal of this development, it was realised that to generalize Artin reciprocity to the non-abelian case, it was essential in fact to seek a new way of expressing Artin L-functions. The contemporary formulation of this ambition is by means of the Langlands program: in which grounds are given for believing Artin L-functions are also L-functions of automorphic representations.[4] As of the early twenty-first century, this is the formulation of the notion of non-abelian class field theory that has widest expert acceptance.[5]
154.2 Notes [1] The problem of creating non-Abelian class field theory for normal extensions with non-Abelian Galois group remains. From Kuz'min, L.V. (2001), “Class field theory”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4. [2] On the statistical level, the classical result on primes in arithmetic progressions of Dirichlet generalises to Chebotaryov’s density theorem; what is asked for is a generalisation, of the same scope of quadratic reciprocity. [3] In today’s terminology, that is the second inequality. See class formation for a contemporary presentation. [4] James W. Cogdell, Functoriality, Converse Theorems and Applications (PDF) states that Functoriality itself is a manifestation of Langlands’ vision of a non-abelian class field theory.
467
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CHAPTER 154. NON-ABELIAN CLASS FIELD THEORY
[5] The matter of reciprocity laws and symbols for non-Abelian field extensions more properly fits into non-Abelian class field theory and the Langlands program: from Hazewinkel, M. (2001), “Hilbert problems”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 155
Norm form In mathematics, a norm form is a homogeneous form in n variables constructed from the field norm of a field extension L/K of degree n.[1] That is, writing N for the norm mapping to K, and selecting a basis e1 , ..., en for L as a vector space over K, the form is given by N(x1 e1 + ... + xnen) in variables x1 , ..., xn. In number theory norm forms are studied as Diophantine equations, where they generalize, for example, the Pell equation.[2] For this application the field K is usually the rational number field, the field L is an algebraic number field, and the basis is taken of some order in the ring of integers OL of L.
155.1 References [1] Lekkerkerker, C. G. (1969), Geometry of numbers, Bibliotheca Mathematica 8, Amsterdam: North-Holland Publishing Co., p. 29, MR 0271032. [2] Bombieri, Enrico; Gubler, Walter (2006), Heights in Diophantine geometry, New Mathematical Monographs 4, Cambridge University Press, Cambridge, pp. 190–191, doi:10.1017/CBO9780511542879, ISBN 978-0-521-84615-8, MR 2216774.
469
Chapter 156
Norm group In number theory, a norm group is a group of the form NL/K (L× ) where L/K is a finite abelian extension of nonarchimedean local fields. One of the main theorems in local class field theory states that the norm groups in K × are precisely the open subgroups of K × of finite index.
156.1 See also • Takagi existence theorem
156.2 References • J.S. Milne, Class field theory. Version 4.01.
470
Chapter 157
Normal basis In mathematics, a normal basis in field theory is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory the study of the more refined question of the existence of a normal integral basis is part of Galois module theory. In the case of finite fields, this means that each of the basis elements is related to any one of them by applying the Frobenius p-th power mapping repeatedly, where p is the characteristic of the field. Let GF(pm ) be a field with pm elements, and β an element of it such that the m elements
2
{β, β p , β p , . . . , β p
m−1
}
are linearly independent. Then this set forms a normal basis for GF(pm ) over GF(p).
157.1 Usage This basis is frequently used in cryptographic applications that are based on the discrete logarithm problem such as elliptic curve cryptography. Hardware implementations of normal basis arithmetic typically have far less power consumption than other bases. 2
When representing elements as a binary string (e.g. in GF(23 ) the most significant bit represents β2 =β4 , the middle 1 0 bit represents β2 =β2 , and the least significant bit represents β2 =β), we can square elements by doing a left circular shift (left shifting β4 would give β8 , but since we are working in GF(23 ) this wraps around to β). This makes the normal basis especially attractive for cryptosystems that utilize frequent squaring.
157.2 Primitive normal basis A primitive normal basis of an extension of finite fields E/F is a normal basis for E/F which is generated by a primitive element of E. Lenstra and Schoof (1987) proved that every finite field extension possesses a primitive normal basis, the case when F is a prime field having been settled by Harold Davenport.
157.3 Free elements If E/F is a Galois extension with group G and x in E generates a normal basis then x is free in E/F. If x has the property that for every subgroup H of G, with fixed field H°, x is free for E/H°, then x is said to be completely free in E/F. Every Galois extension has a completely free element.[1] 471
472
CHAPTER 157. NORMAL BASIS
157.4 See also • Dual basis in a field extension • Polynomial basis • Zech’s logarithms for reducing high-order polynomials to those within the field
157.5 References [1] Dirk Hachenberger, Completely free elements, in Cohen & Niederreiter (1996) pp.97-107 Zbl 0864.11066
• Cohen, S.; Niederreiter, H., eds. (1996). Finite Fields and Applications. Proceedings of the 3rd international conference, Glasgow, UK, July 11–14, 1995. London Mathematical Society Lecture Note Series 233. Cambridge University Press. ISBN 0-521-56736-X. Zbl 0851.00052. • Lenstra, H.W., jr; Schoof, R.J. (1987). “Primitive normal bases for finite fields”. Mathematics of Computation 48: 217–231. doi:10.2307/2007886. Zbl 0615.12023. • Menezes, Alfred J., ed. (1993). Applications of finite fields. The Kluwer International Series in Engineering and Computer Science 199. Boston: Kluwer Academic Publishers. ISBN 0792392825. Zbl 0779.11059. • Stewart, Ian (1990). Galois theory (3rd ed.). CRC Press. ISBN 978-0-412-34550-0. Zbl 1049.12001.
Chapter 158
Normal extension In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]. Bourbaki calls such an extension a quasi-Galois extension.
158.1 Equivalent properties and examples The normality of L/K is equivalent to either of the following properties. Let K a be an algebraic closure of K containing L. • Every embedding σ of L in K a that restricts to the identity on K, satisfies σ(L) = L. In other words, σ is an automorphism of L over K. • Every irreducible polynomial in K[X] that has one root in L, has all of its roots in L, that is, it decomposes into linear factors in L[X]. (One says that the polynomial splits in L.) If L is a finite extension of K that is separable (for example, this is automatically satisfied if K is finite or has characteristic zero) then the following property is also equivalent: • There exists an irreducible polynomial whose roots, together with the elements of K, generate L. (One says that L is the splitting field for the polynomial.) √ √ For example, Q( 2) is a normal extension of Q , since it is a splitting field of x2 − 2. On the other hand, Q( 3 2) is √ not a normal extension of Q since the irreducible polynomial x3 − 2 has one root in it (namely, 3 2 ), but not all of them (it does not have the non-real cubic roots of 2). √ The fact that Q( 3 2) is not a normal extension of Q can also be seen using √ the first of the three properties above. The field A of algebraic numbers is an algebraic closure of Q containing Q( 3 2) . On the other hand √ √ √ 3 3 3 Q( 2) = {a + b 2 + c 4 ∈ A | a, b, c ∈ Q} and, if ω is one of the two non-real cubic roots of 2, then the map
σ:
√ 3 2) √ −→ Q( √ √A √ a + b 3 2 + c 3 4 7→ a + bω 3 2 + cω 2 3 4
√ √ is an embedding of Q( 3 2) in A whose restriction to Q is the identity. However, σ is not an automorphism of Q( 3 2) . √ For any prime p, the extension Q( p 2, ζp ) is normal√of degree p(p − 1). It is a splitting field of xp√− 2. Here ζp denotes any pth primitive root of unity. The field Q( 3 2, ζ3 ) is the normal closure (see below) of Q( 3 2) . 473
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CHAPTER 158. NORMAL EXTENSION
158.2 Other properties Let L be an extension of a field K. Then: • If L is a normal extension of K and if E is an intermediate extension (i.e., L ⊃ E ⊃ K), then L is a normal extension of E. • If E and F are normal extensions of K contained in L, then the compositum EF and E ∩ F are also normal extensions of K.
158.3 Normal closure If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is a normal extension of K. Furthermore, up to isomorphism there is only one such extension which is minimal, i.e. such that the only subfield of M which contains L and which is a normal extension of K is M itself. This extension is called the normal closure of the extension L of K. If L is a finite extension of K, then its normal closure is also a finite extension.
158.4 See also • Galois extension • Normal basis
158.5 References • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: SpringerVerlag, ISBN 978-0-387-95385-4, MR 1878556 • Jacobson, Nathan (1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9, MR 1009787
Chapter 159
p-adic Hodge theory In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields[1] with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.
159.1 General classification of p-adic representations Let K be a local field of residue field k of characteristic p. In this article, a p-adic representation of K (or of GK, the absolute Galois group of K) will be a continuous representation ρ : GK→ GL(V) where V is a finite-dimensional vector space over Qp. The collection of all p-adic representations of K form an abelian category denoted RepQp (K) in this article. p-adic Hodge theory provides subcollections of p-adic representations based on how nice they are, and also provides faithful functors to categories of linear algebraic objects that are easier to study. The basic classification is as follows:[2]
Repcris (K) ⊊ Repst (K) ⊊ RepdR (K) ⊊ RepHT (K) ⊊ RepQp (K) where each collection is a full subcategory properly contained in the next. In order, these are the categories of crystalline representations, semistable representations, de Rham representations, Hodge–Tate representations, and all p-adic representations. In addition, two other categories of representations can be introduced, the potentially crystalline representations Rep ᵣᵢ (K) and the potentially semistable representations Rep (K). The latter strictly contains the former which in turn generally strictly contains Rep ᵣᵢ (K); additionally, Rep (K) generally strictly contains Rep (K), and is contained in Rep R(K) (with equality when the residue field of K is finite, a statement called the p-adic monodromy theorem).
159.2 Period rings and comparison isomorphisms in arithmetic geometry The general strategy of p-adic Hodge theory, introduced by Fontaine, is to construct certain so-called period rings[3] such as B R, B , B ᵣᵢ , and BHT which have both an action by GK and some linear algebraic structure and to consider so-called Dieudonné modules
DB (V ) = (B ⊗Qp V )GK (where B is a period ring, and V is a p-adic representation) which no longer have a GK-action, but are endowed with linear algebraic structures inherited from the ring B. In particular, they are vector spaces over the fixed field 475
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CHAPTER 159. P-ADIC HODGE THEORY
E := B GK .[4] This construction fits into the formalism of B-admissible representations introduced by Fontaine. For a period ring like the aforementioned ones B∗ (for ∗ = HT, dR, st, cris), the category of p-adic representations Rep∗(K) mentioned above is the category of B∗-admissible ones, i.e. those p-adic representations V for which
dimE DB∗ (V ) = dimQp V or, equivalently, the comparison morphism
αV : B∗ ⊗E DB∗ (V ) −→ B∗ ⊗Qp V is an isomorphism. This formalism (and the name period ring) grew out of a few results and conjectures regarding comparison isomorphisms in arithmetic and complex geometry: • If X is a proper smooth scheme over C, there is a classical comparison isomorphism between the algebraic de Rham cohomology of X over C and the singular cohomology of X(C)
∗ HdR (X/C) ∼ = H ∗ (X(C), Q) ⊗Q C.
This isomorphism can be obtained by considering a pairing obtained by integrating differential forms in the algebraic de Rham cohomology over cycles in the singular cohomology. The result of such an integration is called a period and is generally a complex number. This explains why the singular cohomology must be tensored to C, and from this point of view, C can be said to contain all the periods necessary to compare algebraic de Rham cohomology with singular cohomology, and could hence be called a period ring in this situation. • In the mid sixties, Tate conjectured[5] that a similar isomorphism should hold for proper smooth schemes X over K between algebraic de Rham cohomology and p-adic étale cohomology (the Hodge–Tate conjecture, also called CHT). Specifically, let CK be the completion of an algebraic closure of K, let CK(i) denote CK where the action of GK is via g·z = χ(g)i g·z (where χ is the p-adic cyclotomic character, and i is an integer), and let BHT := ⊕i∈Z CK (i) . Then there is a functorial isomorphism
∗ BHT ⊗K grHdR (X/K) ∼ = BHT ⊗Qp Het∗ (X ×K K, Qp )
of graded vector spaces with GK-action (the de Rham cohomology is equipped with the Hodge filtration, ∗ and grHdR is its associated graded). This conjecture was proved by Gerd Faltings in the late eighties[6] after partial results by several other mathematicians (including Tate himself). • For an abelian variety X with good reduction over a p-adic field K, Alexander Grothendieck reformulated a theorem of Tate’s to say that the crystalline cohomology H 1 (X/W(k)) ⊗ Qp of the special fiber (with the Frobenius endomorphism on this group and the Hodge filtration on this group tensored with K) and the p-adic étale cohomology H 1 (X,Qp) (with the action of the Galois group of K) contained the same information. Both are equivalent to the p-divisible group associated to X, up to isogeny. Grothendieck conjectured that there should be a way to go directly from p-adic étale cohomology to crystalline cohomology (and back), for all varieties with good reduction over p-adic fields.[7] This suggested relation became known as the mysterious functor. To improve the Hodge–Tate conjecture to one involving the de Rham cohomology (not just its associated graded), Fontaine constructed[8] a filtered ring B R whose associated graded is BHT and conjectured[9] the following (called C R) for any smooth proper scheme X over K ∗ BdR ⊗K HdR (X/K) ∼ = BdR ⊗Qp Het∗ (X ×K K, Qp )
159.3. NOTES
477
as filtered vector spaces with GK-action. In this way, B R could be said to contain all (p-adic) periods required to compare algebraic de Rham cohomology with p-adic étale cohomology, just as the complex numbers above were used with the comparison with singular cohomology. This is where B R obtains its name of ring of p-adic periods. Similarly, to formulate a conjecture explaining Grothendieck’s mysterious functor, Fontaine introduced a ring B ᵣᵢ with GK-action, a “Frobenius” φ, and a filtration after extending scalars from K 0 to K. He conjectured[10] the following (called C ᵣᵢ ) for any smooth proper scheme X over K with good reduction ∗ Bcris ⊗K0 HdR (X/K) ∼ = Bcris ⊗Qp Het∗ (X ×K K, Qp ) ∗ as vector spaces with φ-action, GK-action, and filtration after extending scalars to K (here HdR (X/K) is given its structure as a K 0 -vector space with φ-action given by its comparison with crystalline cohomology). Both the C R and the C ᵣᵢ conjectures were proved by Faltings.[11]
Upon comparing these two conjectures with the notion of B∗-admissible representations above, it is seen that if X is a proper smooth scheme over K (with good reduction) and V is the p-adic Galois representation obtained as is its ith p-adic étale cohomology group, then
i DB∗ (V ) = HdR (X/K).
In other words, the Dieudonné modules should be thought of as giving the other cohomologies related to V. In the late eighties, Fontaine and Uwe Jannsen formulated another comparison isomorphism conjecture, C , this time allowing X to have semi-stable reduction. Fontaine constructed[12] a ring B with GK-action, a “Frobenius” φ, a filtration after extending scalars from K 0 to K (and fixing an extension of the p-adic logarithm), and a “monodromy operator” N. When X has semi-stable reduction, the de Rham cohomology can be equipped with the φ-action and a monodromy operator by its comparison with the log-crystalline cohomology first introduced by Osamu Hyodo.[13] The conjecture then states that ∗ Bst ⊗K0 HdR (X/K) ∼ = Bst ⊗Qp Het∗ (X ×K K, Qp )
as vector spaces with φ-action, GK-action, filtration after extending scalars to K, and monodromy operator N. This conjecture was proved in the late nineties by Takeshi Tsuji.[14]
159.3 Notes [1] In this article, a local field is complete discrete valuation field whose residue field is perfect. [2] Fontaine 1994, p. 114 [3] These rings depend on the local field K in question, but this relation is usually dropped from the notation. [4] For B = BHT, B R, B , and B ᵣᵢ , B GK is K, K, K 0 , and K 0 , respectively, where K 0 = Frac(W(k)), the fraction field of the Witt vectors of k. [5] See Serre 1967 [6] Faltings 1988 [7] Grothendieck 1971, p. 435 [8] Fontaine 1982 [9] Fontaine 1982, Conjecture A.6 [10] Fontaine 1982, Conjecture A.11 [11] Faltings 1989 [12] Fontaine 1994, Exposé II, section 3 [13] Hyodo 1991 [14] Tsuji 1999
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CHAPTER 159. P-ADIC HODGE THEORY
159.4 References 159.4.1
Primary sources
• Faltings, Gerd (1988), "p-adic Hodge theory”, Journal of the American Mathematical Society 1 (1): 255–299, doi:10.2307/1990970, MR 0924705 • Faltings, Gerd, “Crystalline cohomology and p-adic Galois representations”, in Igusa, Jun-Ichi, Algebraic analysis, geometry, and number theory, Baltimore, MD: Johns Hopkins University Press, pp. 25–80, ISBN 978-08018-3841-5, MR 1463696 • Fontaine, Jean-Marc (1982), “Sur certains types de représentations p-adiques du groupe de Galois d'un corps local; construction d'un anneau de Barsotti–Tate”, Annals of Mathematics 115 (3): 529–577, doi:10.2307/2007012, MR 0657238 • Grothendieck, Alexander (1971), “Groupes de Barsotti–Tate et cristaux”, Actes du Congrès International des Mathématiciens (Nice, 1970) 1, pp. 431–436, MR 0578496 • Hyodo, Osamu (1991), “On the de Rham–Witt complex attached to a semi-stable family”, Compositio Mathematica 78 (3): 241–260, MR 1106296 • Serre, Jean-Pierre (1967), “Résumé des cours, 1965–66”, Annuaire du Collège de France, Paris, pp. 49–58 • Tsuji, Takeshi (1999), "p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case”, Inventiones Mathematicae 137 (2): 233–411, doi:10.1007/s002220050330, MR 1705837
159.4.2
Secondary sources
• Berger, Laurent (2004), “An introduction to the theory of p-adic representations”, Geometric aspects of Dwork theory I, Berlin: Walter de Gruyter GmbH & Co. KG, arXiv:math/0210184, ISBN 978-3-11-017478-6, MR 2023292 • Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory, retrieved 2010-02-05 • Fontaine, Jean-Marc, ed. (1994), Périodes p-adiques, Astérisque 223, Paris: Société Mathématique de France, MR 1293969 • Illusie, Luc (1990), “Cohomologie de de Rham et cohomologie étale p-adique (d'après G. Faltings, J.-M. Fontaine et al.) Exp. 726”, Séminaire Bourbaki. Vol. 1989/90. Exposés 715–729, Astérisque, 189–190, Paris: Société Mathématique de France, pp. 325–374, MR 1099881
Chapter 160
p-adic number
The 3-adic integers, with selected corresponding characters on their Pontryagin dual group
In mathematics the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of “closeness” or absolute value. In particular, 479
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p-adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of p – the higher the power the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat’s Last Theorem by Andrew Wiles.[1] p-adic numbers were first described by Kurt Hensel in 1897,[2] though with hindsight some of Kummer’s earlier work can be interpreted as implicitly using p-adic numbers.[3] The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus. More formally, for a given prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is itself derived from the p-adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure which gives the p-adic number systems their power and utility. The p in p-adic is a variable and may be replaced with a prime (yielding, for instance, “the 2-adic numbers”) or another placeholder variable (for expressions such as “the ℓ-adic numbers”). The “adic” of "p-adic” comes from the ending found in words such as dyadic or triadic, and the p means a prime number.
160.1 Introduction This section is an informal introduction to p-adic numbers, using examples from the ring of 10-adic (decadic) numbers. Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals. The decadic numbers are generally not used in mathematics: since 10 is not prime, the decadics are not a field. More formal constructions and properties are given below. In the standard decimal representation, almost all[4] real numbers do not have a terminating decimal representation. For example, 1/3 is represented as a non-terminating decimal as follows 1 = 0.333333 . . . . 3 Informally, non-terminating decimals are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. If two decimal expansions differ only after the 10th decimal place, they are quite close to one another; and if they differ only after the 20th decimal place, they are even closer. 10-adic numbers use a similar non-terminating expansion, but with a different concept of “closeness”. Whereas two decimal expansions are close to one another if their difference is a large negative power of 10, two 10-adic expansions are close if their difference is a large positive power of 10. Thus 3333 and 4333, which differ by 103 , are close in the 10-adic world, and 33333333 and 43333333 are even closer, differing by 107 . More precisely, a rational number r can be expressed as 10e ·p/q, where p and q are positive integers and q is relatively prime to p and to 10. For each r ≠ 0 there exists the maximal e such that this representation is possible. Let the 10-adic norm of r to be |r|10 =
1 10e
|0|10 = 0. Closeness in any number system is defined by a metric. Using the 10-adic metric the distance between numbers x and y is given by |x − y|10 . An interesting consequence of the 10-adic metric (or of a p-adic metric) is that there is no longer a need for the negative sign. As an example, by examining the following sequence we can see how unsigned 10-adics can get progressively closer and closer to the number −1: 9 = −1 + 10 so |9 − (−1)|10 =
1 10
99 = −1 + 10 so |99 − (−1)|10 = 2
. 1 100
999 = −1 + 10 so |999 − (−1)|10 = 3
.
1 1000
9999 = −1 + 10 so |9999 − (−1)|10 = 4
.
1 10000
.
160.1. INTRODUCTION
481
and taking this sequence to its limit, we can say that the 10-adic expansion of −1 is
. . . 9999 = −1. In this notation, 10-adic expansions can be extended indefinitely to the left, in contrast to decimal expansions, which can be extended indefinitely to the right. Note that this is not the only way to write p-adic numbers – for alternatives see the Notation section below. More formally, a 10-adic number can be defined as ∞ ∑
ai 10i
i=n
where each of the ai is a digit taken from the set {0, 1, … , 9} and the initial index n may be positive, negative or 0, but must be finite. From this definition, it is clear that positive integers and positive rational numbers with terminating decimal expansions will have terminating 10-adic expansions that are identical to their decimal expansions. Other numbers may have non-terminating 10-adic expansions. It is possible to define addition, subtraction, and multiplication on 10-adic numbers in a consistent way, so that the 10-adic numbers form a commutative ring. We can create 10-adic expansions for negative numbers as follows
−100 = −1 × 100 = . . . 9999 × 100 = . . . 9900 ⇒ −35 = −100 + 65 = . . . 9900 + 65 = . . . 9965 ( ) 1 −35 . . . 9965 ⇒− 3+ = = = . . . 9996.5 2 10 10 and fractions which have non-terminating decimal expansions also have non-terminating 10-adic expansions. For example
106 − 1 1012 − 1 1018 − 1 = 142857; = 142857142857; = 142857142857142857 7 7 7 ⇒−
1 = . . . 142857142857142857 7
⇒−
6 = . . . 142857142857142857 × 6 = . . . 857142857142857142 7
⇒
1 6 = − + 1 = . . . 857142857142857143. 7 7
Generalizing the last example, we can find a 10-adic expansion with no digits to the right of the decimal point for any rational number p⁄q such that q is co-prime to 10; Euler’s theorem guarantees that if q is co-prime to 10, then there is an n such that 10n − 1 is a multiple of q. The other rational numbers can be expressed as 10-adic numbers with some digits after the decimal point. As noted above, 10-adic numbers have a major drawback. It is possible to find pairs of non-zero 10-adic numbers (having an infinite number of digits, and thus not rational) whose product is 0.[5] This means that 10-adic numbers do not always have multiplicative inverses i.e. valid reciprocals, which in turn implies that though 10-adic numbers form a ring they do not form a field, a deficiency that makes them much less useful as an analytical tool. Another way of saying this is that the ring of 10-adic numbers is not an integral domain because they contain zero divisors. The reason for this property turns out to be that 10 is a composite number which is not a power of a prime. This problem is simply avoided by using a prime number p as the base of the number system instead of 10 and indeed for this reason p in p-adic is usually taken to be prime.
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160.2
p-adic expansions
When dealing with natural numbers, if we take p to be a fixed prime number, then any positive integer can be written as a base p expansion in the form n ∑
ai p i
i=0
where the ai are integers in {0, … , p − 1}.[6] For example, the binary expansion of 35 is 1·25 + 0·24 + 0·23 + 0·22 + 1·21 + 1·20 , often written in the shorthand notation 1000112 . The familiar approach to extending this description to the larger domain of the rationals (and, ultimately, to the reals) is to use sums of the form: n ∑
±
ai pi .
i=−∞
A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...5 . In this formulation, the integers are precisely those numbers for which ai = 0 for all i < 0. With p-adic numbers, on the other hand, we choose to extend the base p expansions in a different way. Unlike traditional integers, where the magnitude is determined by how far they are from zero, the “size” of p-adic numbers is determined by the p-adic Norm, where high positive powers of p are relatively small compared to high negative powers of p. Consider infinite sums of the form: ∞ ∑
ai p i
i=k
where k is some (not necessarily positive) integer, and each coefficient ai can be called a p-adic digit.[7] With this approach we obtain the p-adic expansions of the p-adic numbers. Those p-adic numbers for which ai = 0 for all i < 0 are also called the p-adic integers. As opposed to real number expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p, p-adic numbers may expand to the left forever, a property that can often be true for the p-adic integers. For example, consider the p-adic expansion of 1/3 in base 5. It can be shown to be …13131325 , i.e., the limit of the sequence 25 , 325 , 1325 , 31325 , 131325 , 3131325 , 13131325 , … : 52 − 1 445 54 − 1 44445 = = 135 ; = = 13135 3 3 3 3 1 ⇒ − = . . . 13135 3 2 ⇒ − = . . . 13135 × 2 = . . . 31315 3 1 2 ⇒ = − + 1 = . . . 31325 . 3 3 Multiplying this infinite sum by 3 in base 5 gives …00000015 . As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 satisfies the definition of being a p-adic integer in base 5. More formally, the p-adic expansions can be used to define the field Qp of p-adic numbers while the p-adic integers form a subring of Qp, denoted Zp. (Not to be confused with the ring of integers modulo p which is also sometimes written Zp. To avoid ambiguity, Z/pZ or Z/(p) are often used to represent the integers modulo p.) While it is possible to use the approach above to define p-adic numbers and explore their properties, just as in the case of real numbers other approaches are generally preferred. Hence we want to define a notion of infinite sum which makes these expressions meaningful, and this is most easily accomplished by the introduction of the p-adic metric. Two different but equivalent solutions to this problem are presented in the Constructions section below.
160.3. NOTATION
483
160.3 Notation There are several different conventions for writing p-adic expansions. So far this article has used a notation for p-adic expansions in which powers of p increase from right to left. With this right-to-left notation the 3-adic expansion of 1 ⁄5 , for example, is written as 1 = . . . 1210121023 . 5 When performing arithmetic in this notation, digits are carried to the left. It is also possible to write p-adic expansions so that the powers of p increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of 1 ⁄5 is 1 1 = 2.01210121 . . .3 or = 20.1210121 . . .3 . 5 15 p-adic expansions may be written with other sets of digits instead of {0, 1, …, p − 1}. For example, the 3-adic expansion of 1 /5 can be written using balanced ternary digits {1,0,1} as 1 = . . . 1111111111113 . 5 In fact any set of p integers which are in distinct residue classes modulo p may be used as p-adic digits. In number theory, Teichmüller representatives are sometimes used as digits.[8]
160.4 Constructions 160.4.1
Analytic approach
See also: p-adic order The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; this allows us to, for example, write 1 as 1.000… = 0.999… . The definition of a Cauchy sequence relies on the metric chosen, though, so if we choose a different one, we can construct numbers other than the real numbers. The usual metric which yields the real numbers is called the Euclidean metric. For a given prime p, we define the p-adic absolute value in Q as follows: for any non-zero rational number x, there is a unique integer n allowing us to write x = pn (a/b), where neither of the integers a and b is divisible by p. Unless the numerator or denominator of x in lowest terms contains p as a factor, n will be 0. Now define |x|p = p−n . We also define |0|p = 0. For example with x = 63/550 = 2−1 ·32 ·5−2 ·7·11−1
|x|2 = 2 |x|3 = 1/9 |x|5 = 25 |x|7 = 1/7 |x|11 = 11 |x|prime other any = 1. This definition of |x|p has the effect that high powers of p become “small”. By the fundamental theorem of arithmetic, for a given non-zero rational number x there is a unique finite set of distinct primes p1 , . . . , pr and a corresponding sequence of non-zero integers a1 , . . . , ar such that:
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CHAPTER 160. P-ADIC NUMBER
p=3 1
19
10
4 13
−2
−3
7
6 16
15
18 9
11
0
12
2
3 20
21 17 8
14 −1
5
Similar picture for p = 3 (click to enlarge) shows three closed balls of radius 1/3, where each consists of 3 balls of radius 1/9
|x| = pa1 1 . . . par r . i It then follows that |x|pi = p−a for all 1 ≤ i ≤ r , and |x|p = 1 for any other prime p ∈ / {p1 , . . . , pr }. i
It is a theorem of Ostrowski that each absolute value on Q is equivalent either to the Euclidean absolute value, the trivial absolute value, or to one of the p-adic absolute values for some prime p. So the only norms on Q modulo equivalence are the absolute value, the trivial absolute value and the p-adic absolute value which means that there are only as many completions (with respect to a norm) of Q. The p-adic absolute value defines a metric dp on Q by setting
dp (x, y) = |x − y|p The field Qp of p-adic numbers can then be defined as the completion of the metric space (Q, dp); its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains Q. It can be shown that in Qp, every element x may be written in a unique way as
160.5. PROPERTIES
∞ ∑
485
ai p i
i=k
where k is some integer such that ak ≠ 0 and each ai is in {0, …, p − 1 }. This series converges to x with respect to the metric dp. With this absolute value, the field Qp is a local field.
160.4.2
Algebraic approach
In the algebraic approach, we first define the ring of p-adic integers, and then construct the field of fractions of this ring to get the field of p-adic numbers. We start with the inverse limit of the rings Z/pn Z (see modular arithmetic): a p-adic integer is then a sequence (an)n≥₁ such that an is in Z/pn Z, and if n ≤ m, then an ≡ am (mod pn ). Every natural number m defines such a sequence (an) by an = m mod pn and can therefore be regarded as a p-adic integer. For example, in this case 35 as a 2-adic integer would be written as the sequence (1, 3, 3, 3, 3, 35, 35, 35, …). The operators of the ring amount to pointwise addition and multiplication of such sequences. This is well defined because addition and multiplication commute with the “mod” operator; see modular arithmetic. Moreover, every sequence (an) where the first element is not 0 has an inverse. In that case, for every n, an and p are coprime, and so an and pn are relatively prime. Therefore, each an has an inverse mod pn , and the sequence of these inverses, (bn), is the sought inverse of (an). For example, consider the p-adic integer corresponding to the natural number 7; as a 2-adic number, it would be written (1, 3, 7, 7, 7, 7, 7, ...). This object’s inverse would be written as an ever-increasing sequence that begins (1, 3, 7, 7, 23, 55, 55, 183, 439, 439, 1463 ...). Naturally, this 2-adic integer has no corresponding natural number. Every such sequence can alternatively be written as a series. For instance, in the 3-adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2·3 + 0·32 + 1·33 + 0·34 + ... The partial sums of this latter series are the elements of the given sequence. The ring of p-adic integers has no zero divisors, so we can take the field of fractions to get the field Qp of p-adic numbers. Note that in this field of fractions, every non-integer p-adic number can be uniquely written as p−n u with a natural number n and a unit in the p-adic integers u. This means that Qp = Quot (Zp ) ∼ = (pN )−1 Zp . Note that S −1 A, where S = pN = {pn : n ∈ N} is a multiplicative subset (contains the unit and closed under multiplication) of a commutative ring with unit A , is an algebraic construction called the ring of fractions or localization of A by S .
160.5 Properties 160.5.1 Cardinality Zp is the inverse limit of the finite rings Z/pk Z, but is nonetheless uncountable,[9] and has the cardinality of the continuum. Accordingly, the field Qp is uncountable. The endomorphism ring of the Prüfer p-group of rank n, denoted Z(p∞ )n , is the ring of n × n matrices over Zp; this is sometimes referred to as the Tate module.
160.5.2
Topology
Define a topology on Zp by taking as a basis of open sets all sets of the form Ua(n) = {n + λpa : λ ∈ Zp}.
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1 2
4
8
16 0
A scheme showing the topology of the dyadic (or indeed p-adic) integers. Each clump is an open set made up of other clumps. The numbers in the left-most quarter (containing 1) are all the odd numbers. The next group to the right is the even numbers not divisible by 4.
where a is a non-negative integer and n is an integer in [1, pa ]. For example, in the dyadic integers, U1 (1) is the set of odd numbers. Uₐ(n) is the set of all p-adic integers whose difference from n has p-adic absolute value less than p1−a . Then Zp is a compactification of Z, under the derived topology (it is not a compactification of Z with its usual discrete topology). The relative topology on Z as a subset of Zp is called the p-adic topology on Z. The topology of Zp is that of a Cantor set.[10] For instance, we can make a continuous 1-to-1 mapping between the dyadic integers and the Cantor set expressed in base 3 by mapping · · · d2 d1 d0 in Z2 to 0.e0 e1 e2 · · ·3 in C, where en = 2dn . . Using a different mapping, in which the integers go to just part of the Cantor set, one can show that the topology of Qp is that of a Cantor set minus a point (such as the right-most point).[11] In particular, Zp is compact while Qp is not; it is only locally compact. As metric spaces, both Zp and Qp are complete.[12]
160.5.3
Metric completions and algebraic closures
Qp contains Q and is a field of characteristic 0. This field cannot be turned into an ordered field. R has only a single proper algebraic extension: C; in other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of Qp, denoted Qp, has infinite degree,[13] i.e. Qp has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the p-adic valuation to Qp, the latter is not (metrically) complete.[14][15] Its (metric) completion is called Cp or Ωp.[15][16] Here an end is reached, as Cp is algebraically closed.[15][17] However unlike C this field is not locally compact.[16] Cp and C are isomorphic as fields, so we may regard Cp as C endowed with an exotic metric. It should be noted that the proof of existence of such a field isomorphism relies on the axiom of choice, and does not provide an explicit example of such an isomorphism. If K is a finite Galois extension of Qp, the Galois group Gal(K/Qp) is solvable. Thus, the Galois group Gal(Qp/Qp) is prosolvable.
160.6. RATIONAL ARITHMETIC
160.5.4
487
Multiplicative group of Qp
Qp contains the n-th cyclotomic field (n > 2) if and only if n | p − 1.[18] For instance, the n-th cyclotomic field is a subfield of Q13 if and only if n = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicative p-torsion in Qp, if p > 2. Also, −1 is the only non-trivial torsion element in Q2 . Given a natural number k, the index of the multiplicative group of the k-th powers of the non-zero elements of Qp in Q× p is finite. The number e, defined as the sum of reciprocals of factorials, is not a member of any p-adic field; but e p ∈ Qp (p ≠ 2). For p = 2 one must take at least the fourth power.[19] (Thus a number with similar properties as e – namely a p-th root of e p – is a member of Qp for all p.)
160.5.5
Analysis on Qp
The only real functions whose derivative is zero are the constant functions. This is not true over Qp.[20] For instance, the function f : Qp → { Qp |x|−2 p f (x) = 0
x ̸= 0 x=0
has zero derivative everywhere but is not even locally constant at 0. If we let R be denoted Q∞, then given any elements r∞, r2 , r3 , r5 , r7 , ... where rp ∈ Qp, it is possible to find a sequence (xn) in Q such that for all p (including ∞), the limit of xn in Qp is rp.
160.6 Rational arithmetic Eric Hehner and Nigel Horspool proposed in 1979 the use of a p-adic representation for rational numbers on computers[21] called Quote notation. The primary advantage of such a representation is that addition, subtraction, and multiplication can be done in a straightforward manner analogous to similar methods for binary integers; and division is even simpler, resembling multiplication. However, it has the disadvantage that representations can be much larger than simply storing the numerator and denominator in binary; for example, if 2n − 1 is a Mersenne prime, its reciprocal will require 2n − 1 bits to represent.
160.7 Generalizations and related concepts The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way. This will be described now. Suppose D is a Dedekind domain and E is its field of fractions. Pick a non-zero prime ideal P of D. If x is a non-zero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D. We write ordP(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set |x|P = c− ordP (x) . Completing with respect to this absolute value |.|P yields a field EP, the proper generalization of the field of p-adic numbers to this setting. The choice of c does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when the residue field D/P is finite, to take for c the size of D/P. For example, when E is a number field, Ostrowski’s theorem says that every non-trivial non-Archimedean absolute value on E arises as some |.|P. The remaining non-trivial absolute values on E arise from the different embeddings
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of E into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of E into the fields Cp, thus putting the description of all the non-trivial absolute values of a number field on a common footing.) Often, one needs to simultaneously keep track of all the above-mentioned completions when E is a number field (or more generally a global field), which are seen as encoding “local” information. This is accomplished by adele rings and idele groups.
160.8 Local–global principle Helmut Hasse's local–global principle is said to hold for an equation if it can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p. This principle holds e.g. for equations given by quadratic forms, but fails for higher polynomials in several indeterminates.
160.9 See also • 1 + 2 + 4 + 8 + ... • C-minimal theory • Hensel’s lemma • k-adic notation • Mahler’s theorem • Volkenborn integral • profinite integer
160.10 Notes [1] F. Q. Gouvêa, A Marvelous Proof, The American Mathematical Monthly, Vol. 101, No. 3 (Mar., 1994), pp. 203–222 [2] Hensel, Kurt (1897). "Über eine neue Begründung der Theorie der algebraischen Zahlen”. Jahresbericht der Deutschen Mathematiker-Vereinigung 6 (3): 83–88. [3] Dedekind, Richard; Weber, Heinrich (2012), Theory of Algebraic Functions of One Variable, History of mathematics 39, American Mathematical Society, ISBN 9780821890349. Translation into English by John Stillwell of Theorie der algebraischen Functionen einer Veränderlichen (1882). Translator’s introduction, page 35: “Indeed, with hindsight it becomes apparent that a discrete valuation is behind Kummer’s concept of ideal numbers.” [4] The number of real numbers with terminating decimal representations is countably infinite, while the number of real numbers without such a representation is uncountably infinite. [5] See Gérard Michon’s article at [6] Kelley, John L. (1955). General Topology. New York: Van Nostrand. pp. 22–25. [7] Madore, David. “A first introduction to p-adic numbers” (PDF). [8] Hazewinkel, M., ed. (2009), Handbook of Algebra, Volume 6, Elsevier, p. 342, ISBN 9780080932811. [9] Robert (2000) Chapter 1 Section 1.1 [10] Robert (2000) Chapter 1 Section 2.3 [11] See Talk:p-adic number#Topology. [12] Gouvêa (1997) Corollary 3.3.8 [13] Gouvêa (1997) Corollary 5.3.10
160.11. REFERENCES
489
[14] Gouvêa (1997) Theorem 5.7.4 [15] Cassels (1986) p.149 [16] Koblitz (1980) p.13 [17] Gouvêa (1997) Proposition 5.7.8 [18] Gouvêa (1997) Proposition 3.4.2 [19] Robert (2000) Section 4.1 [20] Robert (2000) Section 5.1 [21] Eric C. R. Hehner, R. Nigel Horspool, A new representation of the rational numbers for fast easy arithmetic. SIAM Journal on Computing 8, 124–134. 1979.
160.11 References • Bachman, George (1964). Introduction to p-adic Numbers and Valuation Theory. Academic Press. ISBN 0-12-070268-1. • Cassels, J. W. S. (1986). Local Fields. London Mathematical Society Student Texts 3. Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006. • Gouvêa, Fernando Q. (1997). p-adic Numbers: An Introduction (2nd ed.). Springer. ISBN 3-540-62911-4. Zbl 0874.11002. • Koblitz, Neal (1980). p-adic analysis: a short course on recent work. London Mathematical Society Lecture Note Series 46. Cambridge University Press. ISBN 0-521-28060-5. Zbl 0439.12011. • Koblitz, Neal (1984). p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics 58 (2nd ed.). Springer. ISBN 0-387-96017-1. • Mahler, Kurt (1981). p-adic numbers and their functions. Cambridge Tracts in Mathematics 76 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-23102-7. Zbl 0444.12013. • Robert, Alain M. (2000). A Course in p-adic Analysis. Springer. ISBN 0-387-98669-3. • Steen, Lynn Arthur (1978). Counterexamples in Topology. Dover. ISBN 0-486-68735-X.
160.12 External links • Weisstein, Eric W., “p-adic Number”, MathWorld. • p-adic integers at PlanetMath.org. • p-adic number at Springer On-line Encyclopaedia of Mathematics • Completion of Algebraic Closure – on-line lecture notes by Brian Conrad • An Introduction to p-adic Numbers and p-adic Analysis - on-line lecture notes by Andrew Baker, 2007 • Efficient p-adic arithmetic (slides) • Introduction to p-adic numbers
Chapter 161
p-adic order In number theory, for a given prime number p, the p-adic order or p-adic additive valuation of a non-zero integer n is the highest exponent ν such that pν divides n. The p-adic valuation of 0 is defined to be ∞ . It is commonly abbreviated ν (n). If n/d is a rational number in lowest terms, so that n and d are relatively prime, then ν (n/d) is equal to ν (n) if p divides n, or -ν (d) if p divides d, or to 0 if it divides neither one. The most important application of the p-adic order is in constructing the field of p-adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.[1]
1 2
4
8
16 0
Distribution of natural numbers by their 2-adic order, labeled with corresponding powers of two in decimal. Zero always has an infinite order
161.1 Definition and Properties 161.1.1
Integers
Let p be a prime in Z. The p-adic order or p-adic valuation for Z is defined as[2] νp : Z → N 490
161.2. P-ADIC NORM
491
{ max{v ∈ N : pv | n} νp (n) = ∞
161.1.2
ifn ̸= 0 ifn = 0
Rational Numbers
The p-adic order can be extended into the rational numbers. We can define[3] νp : Q → Z
νp
(a) b
= νp (a) − νp (b).
Some properties are: νp (m · n) = νp (m) + νp (n) . νp (m+n) ≥ inf{νp (m), νp (n)}. Moreover, if νp (m) ̸= νp (n) , then νp (m+n) = inf{νp (m), νp (n)}. where inf is the Infimum (i.e. the smaller of the two)
161.2
p-adic Norm
From our definition of the p-adic order, we can define the p-adic norm. The p-adic norm of Q is defined as | · |p : Q→R { p−νp (x) |x|p = 0
ifx ̸= 0 ifx = 0
Some properties of the p-adic norm:
|a|p ≥ 0 Non-negativity |a|p = 0 ⇐⇒ a = 0 |ab|p = |a|p |b|p |a + b|p ≤ |a|p + |b|p |a + b|p ≤ max (|a|p , |b|p ) | − a|p = |a|p
Positive-definiteness Multiplicativeness Subadditivity non-archimedean is it Symmetry
A metric space can be formed on the set Q with a (non-archimedean, translation invariant) metric defined by d : Q×Q→R
d(x, y) = |x − y|p .
161.3 See also • Multiplicity (mathematics) • Ostrowski’s theorem
492
CHAPTER 161. P-ADIC ORDER
161.4 References [1] David S. Dummit; Richard M. Foote (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9. [2] Ireland, K., Rosen, M. (2000). A Classical Introduction to Modern Number Theory. Springer-Verlag New York. Inc., p. 3 [3] Khrennikov, A., Nilsson, M. (2004). P-adic Deterministic and Random Dynamics., Kluwer Academic Publishers, p. 9
Chapter 162
p-adically closed field In mathematics, a p-adically closed field is a field that enjoys a closure property that is a close analogue for p-adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965.[1]
162.1 Definition Let K be the field ℚ of rational numbers and v be its usual p-adic valuation (with v(p) = 1 ). If F is a (not necessarily algebraic) extension field of K, itself equipped with a valuation w, we say that (F, w) is formally p-adic when the following conditions are satisfied: • w extends v (that is, w(x) = v(x) for all x in K), • the residue field of w coincides with the residue field of v (the residue field being the quotient of the valuation ring {x ∈ F : w(x) ≥ 0} by its maximal ideal {x ∈ F : w(x) > 0} ), • the smallest positive value of w coincides with the smallest positive value of v (namely 1, since v was assumed to be normalized): in other words, a uniformizer for K remains a uniformizer for F. (Note that the value group of K may be larger than that of F since it may contain infinitely large elements over the latter.) The formally p-adic fields can be viewed as an analogue of the formally real fields. For example, the field ℚ(i) of Gaussian rationals, if equipped with the valuation w given by w(2 + i) = 1 (and w(2 − i) = 0 ) is formally 5-adic (the place v=5 of the rationals splits in two places of the Gaussian rationals since X 2 + 1 factors over the residue field with 5 elements, and w is one of these places). The field of 5-adic numbers (which contains both the rationals and the Gaussian rationals embedded as per the place w) is also formally 5-adic. On the other hand, the field of Gaussian rationals is not formally 3-adic for any valuation, because the only valuation w on it which extends the 3-adic valuation is given by w(3) = 1 and its residue field has 9 elements. When F is formally p-adic but that there does not exist any proper algebraic formally p-adic extension of F, then F is said to be p-adically closed. For example, the field of p-adic numbers is p-adically closed, and so is the algebraic closure of the rationals inside it (the field of p-adic algebraic numbers). If F is p-adically closed, then:[2] • there is a unique valuation w on F which makes F p-adically closed (so it is legitimate to say that F, rather than the pair (F, w) , is p-adically closed), • F is Henselian with respect to this place (that is, its valuation ring is so), • the valuation ring of F is exactly the image of the Kochen operator (see below), • the value group of F is an extension by ℤ (the value group of K) of a divisible group, with the lexicographical order. 493
494
CHAPTER 162. P-ADICALLY CLOSED FIELD
The first statement is an analogue of the fact that the order of a real-closed field is uniquely determined by the algebraic structure. The definitions given above can be copied to a more general context: if K is a field equipped with a valuation v such that • the residue field of K is finite (call q its cardinal and p its characteristic), • the value group of v admits a smallest positive element (call it 1, and say π is a uniformizer, i.e. v(π) = 1 ), • K has finite absolute ramification, i.e., v(p) is finite (that is, a finite multiple of v(π) = 1 ), (these hypotheses are satisfied for the field of rationals, with q=π=p the prime number having valuation 1) then we can speak of formally v-adic fields (or p -adic if p is the ideal corresponding to v) and v-adically complete fields.
162.2 The Kochen operator If K is a field equipped with a valuation v satisfying the hypothesis and with the notations introduced in the previous paragraph, define the Kochen operator by:
γ(z) =
1 zq − z π (z q − z)2 − 1
(when z q − z ̸= ±1 ). It is easy to check that γ(z) always has non-negative valuation. The Kochen operator can be thought of as a p-adic (or v-adic) analogue of the square function in the real case. An extension field F of K is formally v-adic if and only if π1 does not belong to the subring generated over the value ring of K by the image of the Kochen operator on F. This is an analogue of the statement (or definition) that a field is formally real when −1 is not a sum of squares.
162.3 First-order theory The first-order theory of p-adically closed fields (here we are restricting ourselves to the p-adic case, i.e., K is the field of rationals and v is the p-adic valuation) is complete and model complete, and if we slightly enrich the language it admits quantifier elimination. Thus, one can define p-adically closed fields as those whose first-order theory is elementarily equivalent to that of Qp .
162.4 References • Ax, James; Kochen, Simon (1965). “Diophantine problems over local fields. II. A complete set of axioms for 𝑝-adic number theory”. Amer. J. Math. (The Johns Hopkins University Press) 87 (3): 631–648. doi:10.2307/2373066. JSTOR 2373066. • Kochen, Simon (1969). “Integer valued rational functions over the 𝑝-adic numbers: A 𝑝-adic analogue of the theory of real fields”. Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967). American Mathematical Society. pp. 57–73. • Kuhlmann, F.-V. "𝑝-adically closed field”. Springer Online Reference Works: Encyclopaedia of Mathematics. Springer-Verlag. Retrieved 2009-02-03. • Jarden, Moshe; Roquette, Peter (1980). “The Nullstellensatz over 𝔭-adically closed fields”. J. Math. Soc. Japan 32 (3): 425–460. doi:10.2969/jmsj/03230425.
162.5. NOTES
162.5 Notes [1] Ax & Kochen (1965) [2] Jarden & Roquette (1980), lemma 4.1
495
Chapter 163
Parshin chain In number theory, a Parshin chain is a higher-dimensional analogue of a place of an algebraic number field. They were introduced by Parshin (1978) in order to define an analogue of the idele class group for 2-dimensional schemes. A Parshin chain of dimension s on a scheme is a finite sequence of points p0 , p1 , ..., ps such that pi has dimension i and each point is contained in the closure of the next one.
163.1 References
• Kerz, Moritz (2011), “Ideles in higher dimension”, Mathematical Research Letters 18 (4): 699–713, doi:10.4310/mrl.2011.v18.n4. ISSN 1073-2780, MR 2831836 • Parshin, A. N. (1978), “Abelian coverings of arithmetic schemes”, Doklady Akademii Nauk SSSR 243 (4): 855–858, ISSN 0002-3264, MR 514485
496
Chapter 164
Perfect field In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds: • Every irreducible polynomial over k has distinct roots. • Every irreducible polynomial over k is separable. • Every finite extension of k is separable. • Every algebraic extension of k is separable. • Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power. • Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x→xp is an automorphism of k • The separable closure of k is algebraically closed. • Every reduced commutative k-algebra A is a separable algebra; i.e., A ⊗k F is reduced for every field extension F/k. (see below) Otherwise, k is called imperfect. In particular, all fields of characteristic zero and all finite fields are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above). More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism.[1] (This is equivalent to the above condition “every element of k is a pth power” for integral domains.)
164.1 Examples Examples of perfect fields are: • every field of characteristic zero, e.g. the field of rational numbers or the field of complex numbers; • every finite field, e.g. the field Fp = Z/pZ where p is a prime number; • every algebraically closed field; • the union of a set of perfect fields totally ordered by extension; • fields algebraic over a perfect field. In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic p>0. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is 497
498
CHAPTER 164. PERFECT FIELD
• the field k(X) of all rational functions in an indeterminate X , where k has characteristic p>0 (because X has no p-th root in k(X)).
164.2 Field extension over a perfect field Any finitely generated field extension over a perfect field is separably generated.[2]
164.3 Perfect closure and perfection One of the equivalent conditions says that, in characteristic p, a field adjoined with all pr -th roots (r≥1) is perfect; it −∞ is called the perfect closure of k and usually denoted by k p . The perfect closure can be used in a test for separability. More precisely, a commutative k-algebra A is separable if −∞ and only if A ⊗k k p is reduced.[3] In terms of universal properties, the perfect closure of a ring A of characteristic p is a perfect ring Ap of characteristic p together with a ring homomorphism u : A → Ap such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : Ap → B such that v factors through u (i.e. v = fu). The perfect closure always exists; the proof involves “adjoining p-th roots of elements of A", similar to the case of fields.[4] The perfection of a ring A of characteristic p is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map θ : R(A) → A such that for any perfect ring B of characteristic p equipped with a map φ : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows. Consider the projective system
··· → A → A → A → ··· where the transition maps are the Frobenius endomorphism. The inverse limit of this system is R(A) and consists of sequences (x0 , x1 , ... ) of elements of A such that xpi+1 = xi for all i. The map θ : R(A) → A sends (xi) to x0 .[5]
164.4 See also • p-ring • Quasi-finite field
164.5 Notes [1] Serre 1979, Section II.4 [2] Matsumura, Theorem 26.2 [3] Cohn 2003, Theorem 11.6.10 [4] Bourbaki 2003, Section V.5.1.4, page 111 [5] Brinon & Conrad 2009, section 4.2
164.6 References • Bourbaki, Nicolas (2003), Algebra II, Springer, ISBN 978-3-540-00706-7
164.7. EXTERNAL LINKS
499
• Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved 2010-02-05 • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67 (2 ed.), Springer-Verlag, ISBN 978-0-387-90424-5, MR 554237 • Cohn, P.M. (2003), Basic Algebra: Groups, Rings and Fields • Matsumura, H (2003), Commutative ring theory, Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics 8 (2nd ed.)
164.7 External links • Hazewinkel, Michiel, ed. (2001), “Perfect field”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4
Chapter 165
Perfectoid For the video game characters, see Tiny Toon Adventures: Buster Busts Loose!. In mathematics, perfectoid objects occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p. The notion was introduced by Peter Scholze. A perfectoid field is a complete topological field K whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements. Associated to any perfectoid field K there is another K ♭ of characteristic p where the multiplication may be defined as
K ♭ = lim K . ←− x7→xp The absolute Galois groups of K and K ♭ are isomorphic.
165.1 See also • perfect field
165.2 References • Scholze, Peter (2012). “Perfectoid spaces”. Publ. Math., Inst. Hautes Étud. Sci. 116: 245–313. doi:10.1007/s10240012-0042-x. ISSN 0073-8301. Zbl pre06120994.
165.3 External links • Mathoverflow.net - includes a post by Scholze
500
Chapter 166
Polynomial basis In mathematics, a polynomial basis is basis of a polynomial ring, viewed as a vector space over the field of coefficients, or as a free module over the ring of coefficients. The most common polynomial basis is the monomial basis consisting of all monomials. Other useful polynomial bases are the Bernstein basis and the various sequences of orthogonal polynomials. In the case of a finite extension of a finite fields, polynomial basis may also refer to a basis of the extension of the form
{1, α, . . . , αm−1 } , where α is the root of a primitive polynomial of degree m equal of the degree of the extension).[1] The set of elements of GF(pm ) can then be represented as:
m
{0, 1, α, α2 , . . . , αp
−2
}
using Zech’s logarithms.
166.1 Addition Addition using the polynomial basis is as simple as addition modulo p. For example, in GF(3m ):
(2α2 + 2α + 1) + (2α + 2) = 2α2 + 4α + 3
mod 3 = 2α2 + α
In GF(2m ), addition is especially easy, since addition and subtraction modulo 2 are the same thing (so like terms “cancel out”), and furthermore this operation can be done in hardware using the basic XOR logic gate.
166.2 Multiplication Multiplication of two elements in the polynomial basis can be accomplished in the normal way of multiplication, but there are a number of ways to speed up multiplication, especially in hardware. Using the straightforward method to multiply two elements in GF(pm ) requires up to m2 multiplications in GF(p) and up to m2 − m additions in GF(p). Some of the methods for reducing these values include: • Lookup tables — a prestored table of results; mainly used for small fields, otherwise the table is too large to implement 501
502
CHAPTER 166. POLYNOMIAL BASIS
• The Karatsuba algorithm — repeatedly breaking the multiplication into pieces, decreasing the total number of multiplications but increasing the number of additions. As seen above, addition is very simple, but the overhead in breaking down and recombining the parts make it prohibitive for hardware, although it is often used in software. It can even be used for general multiplication, and is done in many computer algebra systems. • Linear feedback shift register-based multiplication • Subfield computations — breaking the multiplication in GF(pm ) to multiplications in GF(px ) and GF(py ), where x × y = m. This is not frequently used for cryptographic purposes, since some composite degree fields are avoided because of known attacks on them. • Pipelined multipliers — storing intermediate results in buffers so that new values can be loaded into the multiplier faster • Systolic multipliers — using many cells that communicate with neighboring cells only; typically systolic devices are used for computation-intensive operations where input and output sizes are not as important, such as multiplication.
166.3 Squaring Squaring is an important operation because it can be used for general exponentiation as well as inversion of an element. The most basic way to square an element in the polynomial basis would be to apply a chosen multiplication algorithm on an element twice. In general case, there are minor optimizations that can be made, specifically related to the fact that when multiplying an element by itself, all the bits will be the same. In practice, however, the irreducible polynomial for the field is chosen with very few nonzero coefficients which makes squaring in polynomial basis of GF(2m ) much simpler than multiplication.[2]
166.4 Inversion Inversion of elements can be accomplished in many ways, including: • Lookup tables — once again, only for small fields otherwise the table is too large for implementation • Subfield inversion — by solving systems of equations in subfields m
• Repeated square and multiply — for example in GF(2m ), A−1 = A2
−2
• The Extended Euclidean algorithm • The Itoh-Tsujii inversion algorithm
166.5 Usage The polynomial basis is frequently used in cryptographic applications that are based on the discrete logarithm problem such as elliptic curve cryptography. The advantage of the polynomial basis is that multiplication is relatively easy. For contrast, the normal basis is an alternative to the polynomial basis and it has more complex multiplication but squaring is very simple. Hardware implementations of polynomial basis arithmetic usually consume more power than their normal basis counterparts.
166.6 References [1] Roman, Steven (1995). Field Theory. New York: Springer-Verlag. ISBN 0-387-94407-9. [2] Huapeng, Wu (2001). “Selected Areas in Cryptography: 7th Annual International Workshop, SAC 2000, Waterloo, Ontario, Canada, August 14–15, 2000,”. Springer. p. 118. |chapter= ignored (help)
166.7. SEE ALSO
166.7 See also • normal basis • dual basis • change of basis
503
Chapter 167
Power residue symbol In algebraic number theory the n-th power residue symbol (for an integer n > 2) is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher[1] reciprocity laws.[2]
167.1 Background and notation Let k be an algebraic number field with ring of integers Ok that contains a primitive n-th root of unity ζn ∈ Ok . Let p ⊂ Ok be a prime ideal and assume that n and p are coprime (i.e. n ̸∈ p .) The norm of p is defined as the cardinality of the residue class ring Ok /p : is a finite field)
Np = |Ok /p|. (since p is prime this
There is an analogue of Fermat’s theorem in Ok : If α ∈ Ok , α ̸∈ p, then αNp−1 ≡ 1 (mod p). And finally, Np ≡ 1 (mod n). These facts imply that α
Np−1 n
≡ ζns (mod p) is well-defined and congruent to a unique n-th root of unity ζns .
167.2 Definition This root of unity is called the n-th power residue symbol for Ok , and is denoted by ( ) Np−1 α = ζns ≡ α n p n
(mod p).
167.3 Properties The n-th power symbol has properties completely analogous to those of the classical (quadratic) Legendre symbol: 0 ( ) α = 1 p n ζ where ζ n = 1 and ζ ̸= 1
if α ∈ p if α ̸∈ p and there is an η ∈ Ok such that α ≡ η n if α ̸∈ p and there is no such η
In all cases (zero and nonzero) 504
(mod p)
167.4. RELATION TO THE HILBERT SYMBOL
505
( ) Np−1 α ≡α n (mod p). p n ( ) ( ) ( ) α β αβ = p n p n p n ( ) ( ) α β if α ≡ β (mod p) then = p n p n
167.4 Relation to the Hilbert symbol The n-th power residue symbol is related to the Hilbert symbol (·, ·)p for the prime p by ( ) α = (π, α)p p n in the case p coprime to n, where π is any uniformising element for the local field Kp .[3]
167.5 Generalizations The n-th power symbol may be extended to take non-prime ideals or non-zero elements as its “denominator”, in the same way that the Jacobi symbol extends the Legendre symbol. Any ideal a ⊂ Ok is the product of prime ideals, and in one way only: a = p1 p2 . . . pg . The n-th power symbol is extended multiplicatively: ( ) ( ) ( ) ( ) α α α α = ... . a n p1 n p2 n pg n ( ) is defined as If β ∈ Ok is not zero the symbol α β n
(
( ) α β
= n
α (β)
)
, where (β) is the principal ideal generated by β. n
Analogous to the quadratic Jacobi symbol, this symbol is multiplicative in the top and bottom parameters. ( ) • If α ≡ β (mod a) then •
β a
n
•
(α) (α) a n
b n
= n
( ) ( ) α a
( )
α a
( = n
=
αβ a
(α) ab n
β a
. n
) . n
.
Since the symbol is always an n -th root of unity, because of its multiplicativity it is equal to 1 whenever one parameter is an n -th power; the converse is not true. ( ) • If α ≡ η n (mod a) then αa n = 1. ( ) • If αa n ̸= 1 then α is not an n-th power (mod a). ( ) • If αa n = 1 then α may or may not be an n-th power (mod a).
506
CHAPTER 167. POWER RESIDUE SYMBOL
167.6 Power reciprocity law The power reciprocity law, the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols as[4] ( ) ( )−1 α β
n
β α
=
∏
n
p|n∞ (α, β)p
, whenever α and β are coprime.
167.7 See also • Artin symbol • Gauss’s lemma
167.8 Notes [1] Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers. [2] All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2 [3] Neukirch (1999) p. 336 [4] Neukirch (1999) p. 415
167.9 References • Gras, Georges (2003), Class field theory. From theory to practice, Springer Monographs in Mathematics, Berlin: Springer-Verlag, pp. 204–207, ISBN 3-540-44133-6, Zbl 1019.11032 • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer Science+Business Media, ISBN 0-387-97329-X • Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer Science+Business Media, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002 • Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021
Chapter 168
Primary extension In field theory, a branch of algebra, a primary extension L of K is a field extension such that the algebraic closure of K in L is purely inseparable over K.[1]
168.1 Properties • An extension L/K is primary if and only if it is linearly disjoint from the separable closure of K over K.[1] • A subextension of a primary extension is primary.[1] • A primary extension of a primary extension is primary (transitivity).[1] • Any extension of a separably closed field is primary.[1] • An extension is regular if and only if it is separable and primary.[1] • A primary extension of a perfect field is regular.
168.2 References [1] Fried & Jarden (2008) p.44
• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.). Springer-Verlag. pp. 38–44. ISBN 978-3-540-77269-9. Zbl 1145.12001.
507
Chapter 169
Primitive element (finite field) In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element if it is a primitive (q−1) root of unity in GF(q); this means that all the non-zero elements of GF(q) can be written as αi for some (positive) integer i . For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup of order 3 {2,4,1}; however, 3 is a primitive element of GF(7). The minimal polynomial of a primitive element is a primitive polynomial.
169.1 Properties 169.1.1
Number of Primitive Elements
The number of primitive elements in a finite field GF(q) is φ(q - 1), where φ(m) is Euler’s totient function, which counts the number of elements less than or equal to m which are relatively prime to m. This can be proved by using the theorem that the multiplicative group of a finite field GF(q) is cyclic of order q - 1, and the fact that a finite cyclic group of order m contains φ(m) generators.
169.2 See also • Simple extension • Primitive root • Zech’s logarithm
169.3 References • Lidl, Rudolf; Harald Niederreiter (1997). Finite Fields (2nd ed.). Cambridge University Press. ISBN 0-52139231-4.
169.4 External links • Weisstein, Eric W., “Primitive Polynomial”, MathWorld.
508
Chapter 170
Primitive element theorem In field theory, the primitive element theorem or Artin’s theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element, or simple extensions. It says that a finite extension is simple if and only if there are only finitely many intermediate fields. In particular, finite separable extensions are simple.
170.1 Terminology Let E ⊇ F be a finite field extension. An element α ∈ E is said to be a primitive element for E ⊇ F when
E = F (α). In this situation, the extension E ⊇ F is referred to as a simple extension. Then every element x of E can be written in the form
x = fn−1 αn−1 + · · · + f1 α + f0 , where fi ∈ F for all i, and α ∈ E is fixed. That is, if E ⊇ F is separable of degree n, there exists α ∈ E such that the set
{1, α, · · · , αn−1 } is a basis for E as a vector space over F. √ √ For instance, the extensions Q( 2) ⊇ Q and Q(x) ⊇ Q are simple extensions with primitive elements 2 and x, respectively ( Q(x) denotes the field of rational functions in the indeterminate x over Q ).
170.2 Existence statement The interpretation of the theorem changed with the formulation of the theory of Emil Artin, around 1930. From the time of Galois, the role of primitive elements had been to represent a splitting field as generated by a single element. This (arbitrary) choice of such an element was bypassed in Artin’s treatment.[1] At the same time, considerations of construction of such an element receded: the theorem becomes an existence theorem. The following theorem of Artin then takes the place of the classical primitive element theorem. Theorem 509
510
CHAPTER 170. PRIMITIVE ELEMENT THEOREM
Let E ⊇ F be a finite degree field extension. Then E = F (α) for some element α ∈ E if and only if there exist only finitely many intermediate fields K with E ⊇ K ⊇ F . A corollary to the theorem is then the primitive element theorem in the more traditional sense (where separability was usually tacitly assumed): Corollary Let E ⊇ F be a finite degree separable extension. Then E = F (α) for some α ∈ E . The corollary applies to algebraic number fields, i.e. finite extensions of the rational numbers Q, since Q has characteristic 0 and therefore every extension over Q is separable.
170.3 Counterexamples For non-separable extensions, necessarily in characteristic p with p a prime number, then at least when the degree [L : K] is p, L / K has a primitive element, because there are no intermediate subfields. When [L : K] = p2 , there may not be a primitive element (and therefore there are infinitely many intermediate fields). This happens, for example if K is Fp(T, U), the field of rational functions in two indeterminates T and U over the finite field with p elements, and L is obtained from K by adjoining a p-th root of T, and of U. In fact one can see that for any α in L, the element αp lies in K, but a primitive element must have degree p2 over K.
170.4 Constructive results Generally, the set of all primitive elements for a finite separable extension L / K is the complement of a finite collection of proper K-subspaces of L, namely the intermediate fields. This statement says nothing for the case of finite fields, for which there is a computational theory dedicated to finding a generator of the multiplicative group of the field (a cyclic group), which is a fortiori a primitive element. Where K is infinite, a pigeonhole principle proof technique considers the linear subspace generated by two elements and proves that there are only finitely many linear combinations
γ = α + cβ with c in K, that fail to generate the subfield containing both elements. This is almost immediate as a way of showing how Artin’s result implies the classical result, and a bound for the number of exceptional c in terms of the number of intermediate fields results (this number being something that can be bounded itself by Galois theory and a priori). Therefore in this case trial-and-error is a possible practical method to find primitive elements. See the Example.
170.5 Example It is not, for example, immediately obvious that if one adjoins to the field Q of rational numbers roots of both polynomials
x2 − 2 and
x2 − 3,
170.6. SEE ALSO
511
√ √ √ √ say 2 and 3 respectively, to get a field K = Q( 2, 3) of degree 4 over Q , that the extension is simple and there exists a primitive element γ in K so that K = Q(γ) . One can in fact check that with
γ=
√
2+
√
3
√ √ √ √ √ the powers γ i for 0 ≤ i ≤ 3 can be written out as linear combinations of 1, 2 , 3 , and 2√3 = √6 with integer coefficients. Taking these √ as a3 system of linear equations, or by factoring, one can solve for 2 and 3 over Q(γ) (one gets, for instance, 2 = γ −9γ ), which implies that this choice of γ is indeed a primitive element in this 2 example. A simpler√argument, assuming the knowledge of all the subfields as given by Galois theory, is to note the √ √ √ independence√of 1, √2 , 3√, and 2 3 over the rationals; this shows that the subfield generated by γ cannot be that √ generated by 2 or 3 or 2 3 , exhausting all the subfields of degree 2. Therefore it must be the whole field.
170.6 See also • Primitive element (finite field) • The primitive element theorem at mathreference.com • The primitive element theorem at planetmath.org • The primitive element theorem on Ken Brown’s website (pdf file)
170.7 References [1] Israel Kleiner, A History of Abstract Algebra (2007), p. 64.
170.8 Notes
Chapter 171
Primitive polynomial (field theory) For the use of “primitive polynomial” to mean a polynomial without any non-trivial constant divisor, see Primitive polynomial (ring theory). In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite extension field GF(pm ). In other words, a polynomial F (X) with coefficients in GF(p) = Z/pZ is a primitive m polynomial if its degree is m and it has a root α in GF(pm ) such that {0, 1, α, α2 , α3 , . . . , αp −2 } is the entire field GF(pm ). This means also that α is a primitive (pm − 1)-root of unity in GF(pm ).
171.1 Properties Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible. A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by x. Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root). An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n such that F(x) divides xn − 1 is n = pm − 1. Over GF(pm ) there are exactly φ(pm − 1)/m primitive polynomials of degree m, where φ is Euler’s totient function. A primitive polynomial of degree m has m different roots in GF(pm ), which all have order pm − 1. This means that, m if α is such a root, then αp −1 = 1 and αi ̸= 1 for 0 < i < pm − 1.
171.2 Usage 171.2.1
Field element representation
Primitive polynomials are used in the representation of elements of a finite field. If α in GF(pm ) is a root of a primitive polynomial F(x) then since the order of α is pm − 1 that means that all elements of GF(pm ) can be represented as successive powers of α:
GF (pm ) = {0, 1, α, α2 , . . . , αp
m
−2
}.
When these elements are reduced modulo F(x), they provide the polynomial basis representation of all the elements of the field. Since the multiplicative group of a finite field is always a cyclic group, a primitive polynomial f is a polynomial such that x is a generator of the multiplicative group in GF(p)[x]/f(x) 512
171.3. PRIMITIVE TRINOMIALS
171.2.2
513
Pseudo-random bit generation
Primitive polynomials over GF(2), the field with two elements, can be used for pseudorandom bit generation. In fact, every linear feedback shift register with maximum cycle length (which is 2n − 1, where n is the length of the linear feedback shift register) may be built from a primitive polynomial. For example, given the primitive polynomial x10 + x3 + 1, we start with a user-specified 10-bit seed occupying bit positions 1 through 10, starting from the least significant bit. (The seed need not randomly be chosen, but it can be). We then take the 10th and 3rd bits, and create a new 0th bit, so that the xor of the three bits is 0. The seed is then shifted left one position so that the 0th bit moves to position 1. This process can be repeated to generate 210 − 1 = 1023 pseudo-random bits. In general, for a primitive polynomial of degree m over GF(2), this process will generate 2m − 1 pseudo-random bits before repeating the same sequence.
171.2.3
CRC codes
The cyclic redundancy check (CRC) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF(2) and dividing it by a fixed generator polynomial also over GF(2); see Mathematics of CRC. Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of 2n − 1 for a degree n primitive polynomial.
171.3 Primitive trinomials A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms, because they are the simplest and result in the most efficient pseudo-random number generators. A number of results give techniques for locating and testing primitiveness of trinomials. For trinomials over GF(2), there is a simple test: for every r such that 2r − 1 is a Mersenne prime, a trinomial of degree r is primitive if and only if it is irreducible. Recent algorithms invented by Richard Brent have enabled the discovery of primitive trinomials over GF(2) of very large degree, such as x6972593 + x3037958 + 1. This can be used to create a pseudo-random number generator of the huge period 26972593 − 1, or roughly 102098959 .[1]
171.4 References [1] Search for Primitive Trinomials (mod 2)
171.5 External links • Weisstein, Eric W., “Primitive Polynomial”, MathWorld.
Chapter 172
Principalization (algebra) In the mathematical field of algebraic number theory, the concept principalization has its origin in David Hilbert’s 1897 conjecture that all ideals of an algebraic number field, which can always be generated by two algebraic numbers, become principal ideals, generated by a single algebraic number, when they are transferred to the maximal abelian unramified extension field, which was later called the Hilbert class field, of the given base field. More than thirty years later, Philipp Furtwängler succeeded in proving this principal ideal theorem in 1930, after it had been translated from number theory to group theory by E. Artin in 1929, who made use of his general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of Artin transfers of non-abelian groups with derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to A. Scholz and O. Taussky in 1934, who coined the synonym capitulation for principalization. Another independent access to the principalization problem via Galois cohomology of unit groups is also due to Hilbert and goes back to the chapter on cyclic relative extensions of prime degree in his number report 1897, which culminates in the famous Theorem 94.
172.1 Extension of classes Let K be an algebraic number field, called the base field, and let L|K be a field extension of finite degree. Definition. The embedding monomorphism of fractional ideals ιL|K : IK → IL , a 7→ aOL , where OL denotes the ring of integers of L , induces the extension homomorphism of ideal classes jL|K : IK /PK → IL /PL , aPK 7→ (aOL )PL , where PK and PL denote the subgroups of principal ideals. If there exists a non-principal ideal a ∈ IK , with non trivial class aPK ̸= PK , whose extension ideal in L is principal, aOL = AOL for some number A ∈ L , and hence belongs to the trivial class (aOL )PL = PL , then we speak about principalization or capitulation in L|K . In this case, the ideal a and its class aPK are said to principalize or capitulate in L . This phenomenon is described most conveniently by the principalization kernel or capitulation kernel, that is the kernel ker(jL|K ) of the class extension homomorphism. Remark. When F is a Galois extension of K with automorphism group G = Gal(F |K) such that K ≤ L ≤ F is an intermediate field with relative group H = Gal(F |L) ≤ G , more precise statements about the homomorphisms ιL|K and jL|K are possible by using group theory. According to the theories of A. Hurwitz 1895 [1] and D. Hilbert 1897 [2] on the∏decomposition of a prime∏ ideal p ∈ PK in the extension L|K , viewed as a subextension of F |K , we g have pOL = i=1 qi , where the qi = ϱ∈Hτi D ϱ(P) ∈ PL , with 1 ≤ i ≤ g , are the prime ideals lying over p g in L , expressed by a fixed prime ideal P ∈ PF dividing p in F and a double coset decomposition G = ∪˙ i=1 Hτi D of G modulo H and modulo the decomposition group (stabilizer) D = {σ ∈ G | σ(P) = P} of P in G , with a complete system of representatives (τ1 , . . . , τg ) . The order of the decomposition group D is the inertia degree f (P|p) of P over K . ∏g Consequently,∏the ideal embedding is given by ιL|K (p) = pOL = i=1 qi , and the class extension by jL|K (pPK ) = g (pOL )PL = i=1 qi PL . 514
172.2. ARTIN’S RECIPROCITY LAW
515
172.2 Artin’s reciprocity law Let F |K be a Galois extension of algebraic number fields with automorphism group G = Gal(F |K) . Suppose that p ∈ PK is a prime ideal of K which does not divide the relative discriminant d = d(F |K) , and is therefore unramified in F , and let P ∈ PF be a prime ideal of F lying over p . Then, there exists a unique automorphism σ ∈ G such that[ANorm]K|Q (p) ≡ σ(A) (mod P) , for all algebraic integers |K A ∈ OF , which is called the Frobenius automorphism FP := σ of P and generates the cyclic decomposition group DP = ⟨σ⟩ of P . Any [ other ] prime [ ideal ] of F dividing p is of the form τ (P) with some τ ∈ G . Its Frobenius F |K F |K automorphism is given by τ (P) = τ P τ −1 , since τ (A)NormK|Q (p) ≡ (τ στ −1 )(τ (A)) (mod τ (P)) , for all −1 A ∈ OF , and thus its decomposition Dτ (P) ) [ =] τ DP τ is conjugate to DP . In this general situation, the ( group F |K F |K := {τ P τ −1 | τ ∈ G} which associates an entire conjugacy class of Artin symbol is a mapping p 7→ p ( ) automorphisms to any unramified prime ideal p ̸ |d , and we have F p|K = 1 if and only if p splits completely in F .
Now let F |K be an abelian extension, that is, the Galois group G = Gal(F |K) is an abelian group. Then, all conjugate decomposition ( ) [groups ] of prime ideals of F lying over p coincide Dτ (P) =: Dp , for any τ ∈ G , and the
|K Artin symbol F p|K = FP becomes equal to the Frobenius automorphism of any P | p , since ANormK|Q (p) ≡ ( ) F |K (A) (mod p) , for all A ∈ OF . p
By class field theory, [3] the abelian extension F |K uniquely corresponds to an intermediate group SK,f ≤ H ≤ PK (f) between the ray modulo f of K , that is SK,f = {αOK | α ≡ 1 (mod f)} , and the group of principal ideals coprime to f of K , where f = f(F |K) denotes the relative conductor. (Note ( )that p | f if and only if p | d , but F |K f is minimal with this property.) The Artin symbol PK (f) → G, p 7→ , which associates the Frobenius p automorphism of p to each prime ideal p of K which is unramified in F , can be extended by multiplicativity to an ( ) ∏ vp (a) ∏ ( F |K )vp (a) epimorphism IK (f) → G, a = p 7→ F a|K := with kernel H = SK,f · NormF |K (IF (f)) p ) ( , which induces the Artin isomorphism, or Artin map, IK (f)/H → G = Gal(F |K), aH 7→ F a|K of the generalized ideal class group IK (f)/H to the Galois group G , which maps the class aH of a to the Artin symbol ( ) F |K of a . This explicit isomorphism is called the Artin reciprocity law or general reciprocity law. [4] a
172.3 Commutative diagram E. Artin’s translation of the general principalization problem for a number field extension L|K from number theory to group theory is based on the following scenario. Let F |K be a Galois extension of algebraic number fields with automorphism group G = Gal(F |K) . Suppose that p ∈ PK is a prime ideal of K which does not divide the relative discriminant d = d(F |K) , and is therefore unramified in F , and let P ∈ PF be a prime ideal of F lying over p . Assume that K ≤ L ≤ F is an intermediate field with relative group H = Gal(F |L) ≤ G and let K ′ |K , resp. L′ |L , be the maximal abelian subextension of K , resp. L , within F . Then, the corresponding relative groups are the commutator subgroups G′ = Gal(F |K ′ ) ≤ G , resp. H ′ = Gal(F |L′ ) ≤ H . By class field theory, there exist intermediate P)L (d) such that the ( ′ ) groups SK,d ≤ HK ≤ PK (d) and SL,d ≤ H(L ≤ K |K L′ |L ′ ′ Artin maps establish isomorphisms : IK (d)/HK → Gal(K |K) ≃ G/G and ··· : IL (d)/HL → ··· Gal(L′ |L) ≃ H/H ′ .
The class extension homomorphism jL|K and the Artin transfer, more precisely, the induced transfer T˜G,H , are connected by the commutative ( ′ ) (diagram ) in Figure 1 via these Artin isomorphisms, that is, we have equality of two ′ K |K L |L composita T˜G,H ◦ = ··· ◦ jL|K . [5] The justification for this statement consists in analyzing the two ··· paths of composite mappings. [3] On the one hand, the class extension homomorphism j∏ L|K maps the generalized g ideal class pHK of the base field K to the extension class j (pH ) = (pO )H = K L L L|K i=1 qi HL in the field L ( ) , and the Artin isomorphism
L′ |L ···
of the field L maps this product of classes of prime ideals to the product of [ ]f ∏g ( ′ ) ∏g [ F |L ] ∏g F |K i −1 ′ conjugates of Frobenius automorphisms i=1 Lq|L = · H = τ τi · H ′ . Here, i i=1 i=1 τ (P) P i i the double coset decomposition and its representatives were used, in perfect analogy to the last but one section. On
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CHAPTER 172. PRINCIPALIZATION (ALGEBRA)
Figure 1: Commutative diagram connecting the class extension with the Artin transfer.
( ′ ) the other hand, the Artin isomorphism K···|K of the base field K maps the generalized ideal class pHK to the ( ′ ) ( ′ ) [ ] |K Frobenius automorphism K p|K , and the induced Artin transfer maps the symbol K p|K = FP · G′ to the ) [ ] ]f ([ ∏g |K i −1 |K · G′ = i=1 τi FP product T˜G,H FP τi · H ′ . [6] This product expression was the original form of the Artin transfer homomorphism, corresponding to a decomposition of the permutation representation into disjoint cycles.
172.4 Class field tower The commutative diagram in the previous section, which connects the number theoretic class extension homomorphism jL|K with the group theoretic Artin transfer TG,H , enabled Furtwängler to prove the principal ideal theorem by specializing to the situation that L = F 1 (K) is the first Hilbert class field of K , that is the maximal abelian unramified extension of K , and H = G′ is the commutator subgroup of G . More precisely, Furtwängler showed that generally the Artin transfer TG,G′ from a metabelian group G to its derived subgroup G′ maps all elements of G
172.5. GALOIS COHOMOLOGY
517
to the neutral element of G′ . However, the commutative diagram comprises the potential for a lot of more sophisticated applications. In the situation that p is a prime number, F = Fp2 (K) is the second Hilbert p-class field of K , that is the maximal metabelian unramified extension of K of degree a power of p , L varies over the intermediate field between K and its first Hilbert p-class field Fp1 (K) , and H = Gal(Fp2 (K)|L) ≤ G = Gal(Fp2 (K)|K) correspondingly varies over the intermediate groups between G and G′ , computation of all principalization kernels ker(jL|K ) and all p-class groups Clp (L) translates to information on the kernels ker(TG,H ) and targets H/H ′ of the Artin transfers TG,H and permits the exact specification of the second p-class group G = Gal(Fp2 (K)|K) of K via pattern recognition, and frequently even allows to draw conclusions about the entire p-class field tower of K , that is the Galois group Gal(Fp∞ (K)|K) of the maximal unramified pro-p extension Fp∞ (K) of K . These ideas are explicit in the paper of 1934 by A. Scholz and O. Taussky already. [7] At these early stages, pattern recognition consisted of specifying the annihilator ideals, or symbolic orders, and the Schreier relations of metabelian p-groups and subsequently using a uniqueness theorem on group extensions by O. Schreier. [8] Nowadays, we use the p-group generation algorithm of M. F. Newman [9] and E. A. O'Brien [10] for constructing descendant trees of p-groups and searching patterns, defined by kernels and targets of Artin transfers, among the vertices of these trees.
172.5 Galois cohomology In the chapter on cyclic relative extensions of prime degree of his number report 1897, D. Hilbert [2] proves a series of crucial theorems which culminate in Theorem 94, the original germ of class field theory. Today, these theorems can be viewed as the beginning of what is now called Galois cohomology. Hilbert considers a finite relative extension L|K of algebraic number fields with cyclic Galois group G = Gal(L|K) = ⟨σ⟩ generated by an automorphism σ such that σ ℓ = 1 for the relative degree ℓ = [L : K] , which is assumed to be an odd prime. He investigates two endomorphism of the unit group U = UL of the extension field, viewed as a Galois module with respect to the group G , briefly a G -module. The first endomorphism ∆ : U → U, E 7→ E σ−1 := σ(E)/E is the symbolic exponentiation with the difference σ − 1 ∈ Z[G] , and the second endomorphism N : U → ∏ℓ−1 U, E 7→ E TG := i=0 σ i (E) is the algebraic norm mapping, that is the symbolic exponentiation with the trace ∑ℓ−1 i TG = i=0 σ ∈ Z[G] . In fact, the image of the algebraic norm map is contained in the unit group UK of the base field, and thus N (E) = NL|K (E) coincides with the usual arithmetic norm as the product of all conjugates. The composita of the endomorphisms satisfy the relations ∆ ◦ N = 1 and N ◦ ∆ = 1 . Two important cohomology groups can be defined by means of the kernels and images of these endomorphisms. The zeroth cohomology group of G in UL is given by the quotient H 0 (G, UL ) := ker(∆)/im(N ) = UK /NL|K (UL ) consisting of the norm residues of UK , and the minus first cohomology group of G in UL is given by the quotient H −1 (G, UL ) := ker(N )/im(∆) = EL|K /ULσ−1 of the group EL|K = {E ∈ UL | N (E) = 1} of relative units of L|K modulo the subgroup of symbolic powers of units with formal exponent σ − 1 . In his Theorem 92, under the additional assumption that L|K be unramified, Hilbert proves the existence of a relative unit H ∈ EL|K which cannot be expressed as H = σ(E)/E , for any unit E ∈ UL , which means that the minus first cohomology group H −1 (G, UL ) = EL|K /ULσ−1 is non-trivial of order divisible by ℓ . However, with the aid of a completely similar construction, the minus first cohomology group H −1 (G, L× ) = {A ∈ L× | N (A) = 1}/(L× )σ−1 of the G -module L× = L \ {0} , the multiplicative group of the superfield L , can be defined, and Hilbert shows its triviality H −1 (G, L× ) = 1 in Theorem 90. Applied to the particular unit H ∈ EL|K \ ULσ−1 , this ensures the existence of a non-unit A ∈ L× such that H = Aσ−1 , i. e., Aσ = A · H . The non-unit A is generator of an ambiguous principal ideal of L|K , since (AOL )σ = Aσ OL = A·HOL = AOL . However, the underlying ideal j := (AOL ) ∩ OK of the subfield K cannot be principal, because otherwise we had j = βOK , consequently βOL = AOL , and thus A = βE for some unit E ∈ UL . This would imply the contradiction H = Aσ−1 = (βE)σ−1 = E σ−1 , since β σ−1 = 1 . The ideal j has yet another interesting property. The ℓ th power of its extension ideal jℓ OL = (jOL )ℓ = NL|K (AOL ) = NL|K (A)OL coincides with its relative norm and thus, by forming the intersection with OK , turns out to be principal in the base field K already. Eventually, Hilbert is in the position to state his celebrated Theorem 94: If L|K is a cyclic extension of number fields of odd prime degree ℓ with trivial relative discriminant dL|K = OK , that is, an unramified extension, then there exists a non-principal ideal j ∈ IK \ PK of the base field K which becomes principal jOL = AOL ∈ PL in the extension field L , but the ℓ th power of this non-principal ideal is principal jℓ = NL|K (A)OK ∈ PK in the base
518
CHAPTER 172. PRINCIPALIZATION (ALGEBRA)
field K already. Hence, the class number of the base field must be divisible by ℓ and the extension field L can be called a class field of K . Theorem 94 includes the simple inequality # ker(jL|K ) ≥ ℓ = [L : K] for the order of the principalization kernel of the extension L|K . However, an improved estimate by a possibly bigger lower bound can be derived by means of the theorem on the Herbrand quotient [11] h(G, UL ) of the G -module UL , which is given by h(G, UL ) := #H −1 (G, UL )/#H 0 (G, UL ) = (ker(N ) : im(∆))/(ker(∆) : im(N )) = (EL|K : ULσ−1 )/(UK : NL|K (UL )) = [L : K] , where L|K is a (not necessarily unramified) relative extension of odd degree [L : K] (not necessarily a prime) with cyclic Galois group G = Gal(L|K) . With the aid of K. Iwasawa’s isomorphism [12] between the first cohomology group H 1 (G, UL ) of G in UL and the quotient PLG /PK of the group of ambiguous principal ideals of L modulo the group of principal ideals of K , for any Galois extension L|K with automorphism group G = Gal(L|K) , specialized to a cyclic extension with periodic cohomology of length 2 , and observing that PLG = PL ∩ IK consists of extension ideals only when L|K is unramified, we obtain # ker(jL|K ) = #(PL ∩ IK /PK ) = #(PLG /PK ) = #H 1 (G, UL ) = #H −1 (G, UL ) = [L : K] · #H 0 (G, UL ) = [L : K] · (UK : NL|K (UL )) . This relation increases the lower bound by the factor (UK : NL|K (UL )) , the so-called unit norm index.
172.6 History As mentioned in the lead section, several investigators tried to generalize the Hilbert-Artin-Furtwängler principal ideal theorem of 1930 to questions concerning the principalization in intermediate extensions between the base field and its Hilbert cass field. On the one hand, they established general theorems on the principalization over arbitrary number fields, such as Ph. Furtwängler 1932, [13] O. Taussky 1932, [14] O. Taussky 1970, [15] and H. Kisilevsky 1970. [16] On the other hand, they searched for concrete numerical examples of principalization in unramified cyclic extensions of particular kinds of base fields.
172.6.1
Quadratic fields
√ The principalization of 3 -classes of complex quadratic number fields K = Q( d) with 3 -class rank two in unramified cyclic cubic extensions was calculated manually for three discriminants d ∈ {−3299, −4027, −9748} by A. Scholz and O. Taussky [7] in 1934. Since these calculations require composition of binary quadratic forms and explicit knowledge of fundamental systems of units in cubic number fields, which was a very difficult task in 1934, the investigations stayed at rest for half a century until F.-P. Heider and B. Schmithals [17] employed the CDC Cyber 76 computer at the University of Cologne to extend the information concerning principalization to the range −2 · 104 < d < 105 containing 27 relevant discriminants in 1982, thereby providing the first analysis of five real quadratic fields. Two years later, J. R. Brink [18] computed the principalization types of 66 complex quadratic fields. Currently, the most extensive computation of principalization data for all 4596 quadratic fields with discriminants −106 < d < 107 and 3 -class group of type (3, 3) is due to D. C. Mayer in 2010, [19] who used his recently discovered connection between transfer kernels and transfer targets for the design of a new principalization algorithm. [20] The 2 -principalization in unramified quadratic extensions of complex quadratic fields with 2 -class group of type (2, 2) was studied by H. Kisilevsky in 1976. [21] Similar investigations of real quadratic fields were carried out by E. Benjamin and C. Snyder in 1995. [22]
172.6.2
Cubic fields
The 2 -principalization in unramified quadratic extensions of cyclic cubic number fields with 2 -class group of type (2, 2) was investigated by A. Derhem in 1988. [23] Seven years later, M. Ayadi studied the 3 -principalization in unramified cyclic cubic extensions of cyclic cubic fields K ⊂ Q(ζf ) , ζff = 1 , with 3 -class group of type (3, 3) and conductor f divisible by two or three primes. [24]
172.6.3
Sextic fields
In 1992, M. C. Ismaili investigated √ the 3 -principalization in unramified cyclic cubic extensions of the normal closure of pure cubic fields K = Q( 3 D) , in the case that this sextic number field N = K(ζ3 ) , ζ33 = 1 , has a 3 -class group of type (3, 3) . [25]
172.7. SEE ALSO
172.6.4
519
Quartic fields
In 1993, A. Azizi studied the √ 2√-principalization in unramified quadratic extensions of bicyclic biquadratic fields of Dirichlet type K = Q( d, −1) with 2 -class group of type (2, 2) . [26] Most recently, in 2014, A. Zekhnini extended the investigations to Dirichlet fields with 2 -class group of type (2, 2, 2) , [27] thus providing the first examples of 2 -principalization in the two layers of unramified quadratic and biquadratic extensions of quartic fields with class groups of 2 -rank three.
172.7 See also Both, the algebraic, group theoretic access to the principalization problem by Hilbert-Artin-Furtwängler and the arithmetic, cohomological access by Hilbert-Herbrand-Iwasawa are also presented in detail in the two bibles of capitulation by J.-F. Jaulent 1988 [28] and by K. Miyake 1989. [5]
172.8 Secondary sources • Cassels, J.W.S.; Fröhlich, Albrecht, eds. (1967). Algebraic Number Theory. Academic Press. Zbl 0153.07403. • Iwasawa, Kenkichi (1986). Local class field theory. Oxford Mathematical Monographs. Oxford University Press. ISBN 978-0-19-504030-2. MR 863740. Zbl 0604.12014. • Janusz, Gerald J. (1973). Algebraic number fields. Pure and Applied Mathematics 55. Academic Press. p. 142. Zbl 0307.12001. • Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften 322. Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. • Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften 323 (2nd ed.). Springer-Verlag. ISBN 3-540-37888-X. Zbl 1136.11001.
172.9 References [1] Hurwitz, A. (1926). "Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe”. Math. Z. 25: 661–665. doi:10.1007/bf01283860. [2] Hilbert, D. (1897). “Die Theorie der algebraischen Zahlkörper”. Jahresber. Deutsch. Math. Verein. 4: 175–546. [3] Hasse, H. (1930). “Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz”. Jahresber. Deutsch. Math. Verein., Ergänzungsband 6: 1–204. [4] Artin, E. (1927). “Beweis des allgemeinen Reziprozitätsgesetzes”. Abh. Math. Sem. Univ. Hamburg 5: 353–363. [5] Miyake, K. (1989). “Algebraic investigations of Hilbert’s Theorem 94, the principal ideal theorem and the capitulation problem”. Expo. Math. 7: 289–346. [6] Artin, E. (1929). “Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz”. Abh. Math. Sem. Univ. Hamburg 7: 46–51. [7] Scholz, A., Taussky, O. (1934). “Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm”. J. Reine Angew. Math. 171: 19–41. [8] Schreier, O. (1926). "Über die Erweiterung von Gruppen II”. Abh. Math. Sem. Univ. Hamburg 4: 321–346. [9] Newman, M. F. (1977). Determination of groups of prime-power order. pp. 73-84, in: Group Theory, Canberra, 1975, Lecture Notes in Math., Vol. 573, Springer, Berlin. [10] O'Brien, E. A. (1990). “The p-group generation algorithm”. J. Symbolic Comput. 9: 677–698. doi:10.1016/s07477171(08)80082-x. [11] Herbrand, J. (1932). “Sur les théorèmes du genre principal et des idéaux principaux”. Abh. Math. Sem. Univ. Hamburg 9: 84–92. doi:10.1007/bf02940630.
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[12] Iwasawa, K. (1956). “A note on the group of units of an algebraic number field”. J. Math. Pures Appl. 9 (35): 189–192. [13] Furtwängler, Ph. (1932). "Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper”. J. Reine Angew. Math. 167: 379–387. [14] Taussky, O. (1932). "Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper”. J. Reine Angew. Math. 168: 193–210. [15] Taussky, O. (1970). “A remark concerning Hilbert’s Theorem 94”. J. Reine Angew. Math. 239/240: 435–438. [16] Kisilevsky, H. (1970). “Some results related to Hilbert’s Theorem 94”. J. Number Theory 2: 199–206. doi:10.1016/0022314x(70)90020-x. [17] Heider, F.-P., Schmithals, B. (1982). “Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen”. J. Reine Angew. Math. 363: 1–25. [18] Brink, J. R. (1984). The class field tower for imaginary quadratic number fields of type (3,3). Dissertation, Ohio State Univ. [19] Mayer, D. C. (2012). “The second p-class group of a number field”. Int. J. Number Theory 8 (2): 471–505. doi:10.1142/s179304211250025x. [20] Mayer, D. C. (2014). “Principalization algorithm via class group structure”. J. Théor. Nombres Bordeaux 26 (2): 415–464. [21] Kisilevsky, H. (1976). “Number fields with class number congruent to 4 mod 8 and Hilbert’s Theorem 94”. J. Number Theory 8: 271–279. doi:10.1016/0022-314x(76)90004-4. [22] Benjamin, E., Snyder, C. (1995). “Real quadratic number fields with 2-class group of type (2,2)". Math. Scand. 76: 161–178. [23] Derhem, A. (1988). Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques. Thèse de Doctorat, Univ. Laval, Québec. [24] Ayadi, M. (1995). Sur la capitulation de 3-classes d'idéaux d'un corps cubique cyclique. Thèse de Doctorat, Univ. Laval, Québec. [25] Ismaili, M. C. (1992). Sur la capitulation de 3-classes d'idéaux de la clôture normale d'un corps cubique pure. Thèse de Doctorat, Univ. Laval, Québec. √ [26] Azizi, A. (1993). Sur la capitulation de 2-classes d'idéaux de Q( d, i) . Thèse de Doctorat, Univ. Laval, Québec. √ [27] Zekhnini, A. (2014). Capitulation des 2-classes d'idéaux de certains corps de nombres biquadratiques imaginaires Q( d, i) de type (2,2,2). Thèse de Doctorat, Univ. Mohammed Premier, Faculté des Sciences d'Oujda, Maroc. [28] Jaulent, J.-F. (26 February 1988). “L'état actuel du problème de la capitulation”. Séminaire de Théorie des Nombres de Bordeaux 17: 1–33.
Chapter 173
Profinite integer In mathematics, a profinite integer is an element of the ring
b= Z
∏
Zp
p
b = lim Z/n (profinite completion). where p runs over all prime numbers, Zp is the ring of p-adic integers and Z ←− b .[1] Example: Let Fq be the algebraic closure of a finite field Fq of order q. Then Gal(Fq /Fq ) = Z A usual (rational) integer is a profinite integer since there is the canonical injection
b n 7→ (n, n, . . . ). Z ,→ Z, b ⊗Z Q is the ring of finite adeles AQ,f = ∏ ′ Qp of Q where the prime ' means restricted The tensor product Z p product.[2] There is a canonical paring b → U (1), (q, a) 7→ χ(qa) [3] Q/Z × Z b with the where χ is the character of AQ,f induced by Q/Z → U (1), α 7→ e2πiα .[4] The pairing identifies Z Pontrjagin dual of Q/Z .
173.1 See also • supernatural number
173.2 Notes [1] Milne, Ch. I Example A. 5. [2] http://math.stackexchange.com/questions/233136/questions-on-some-maps-involving-rings-of-finite-adeles-and-their-unit-groups [3] Connes–Consani, § 2.4. [4] K. Conrad, The character group of Q
521
522
CHAPTER 173. PROFINITE INTEGER
173.3 References • Connes, Alain; Consani, Caterina (2015). “Geometry of the arithmetic site”. arXiv:1502.05580. • Milne, Class Field Theory
173.4 External links • http://ncatlab.org/nlab/show/profinite+completion+of+the+integers • http://www.noncommutative.org/supernatural-numbers-and-adeles/
Chapter 174
Proofs of Fermat’s theorem on sums of two squares Fermat’s theorem on sums of two squares asserts that an odd prime number p can be expressed as
p = x2 + y 2 with integer x and y if and only if p is congruent to 1 (mod 4). The statement was announced by Fermat in 1640, but he supplied no proof. The “only if” clause is easy: a perfect square is congruent to 0 or 1 modulo 4, hence a sum of two squares is congruent to 0, 1, or 2. An odd prime number is congruent to either 1 or 3 modulo 4, and the second possibility has just been ruled out. The first proof that such a representation exists was given by Leonhard Euler in 1747 and was complicated. Since then, many different proofs have been found. Among them, the proof using Minkowski’s theorem about convex sets[1] and Don Zagier's short proof based on involutions have appeared.
174.1 Euler’s proof by infinite descent Euler succeeded in proving Fermat’s theorem on sums of two squares in 1749, when he was forty-two years old. He communicated this in a letter to Goldbach dated 12 April 1749.[2] The proof relies on infinite descent, and is only briefly sketched in the letter. The full proof consists in five steps and is published in two papers. The first four steps are Propositions 1 to 4 of the first paper[3] and do not correspond exactly to the four steps below. The fifth step below is from the second paper.[4] 1. The product of two numbers, each of which is a sum of two squares, is itself a sum of two squares. This is a well known property, based on the identity
(a2 + b2 )(p2 + q 2 ) = (ap + bq)2 + (aq − bp)2 due to Diophantus of Alexandria. 2. If a number which is a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares. (This is Euler’s first Proposition). Indeed, suppose for example that a2 + b2 is divisible by p2 + q 2 and that this latter is a prime. Then p2 + q 2 divides
(pb − aq)(pb + aq) = p2 b2 − a2 q 2 = p2 (a2 + b2 ) − a2 (p2 + q 2 ). 523
524
CHAPTER 174. PROOFS OF FERMAT’S THEOREM ON SUMS OF TWO SQUARES Since p2 + q 2 is a prime, it divides one of the two factors. Suppose that it divides pb − aq . Since
(a2 + b2 )(p2 + q 2 ) = (ap + bq)2 + (aq − bp)2 (Diophantus identity) it follows that p2 + q 2 must divide (ap + bq)2 . So the equation can be divided by the square of p2 + q 2 . Dividing the expression by (p2 + q 2 )2 yields:
a2 + b2 = p2 + q 2
(
ap + bq p2 + q 2
)2
( +
aq − bp p2 + q 2
)2
and thus expresses the quotient as a sum of two squares, as claimed. If p2 + q 2 divides pb + aq , a similar argument holds by using
(a2 + b2 )(q 2 + p2 ) = (aq + bp)2 + (ap − bq)2 (Diophantus identity). 3. If a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares. (This is Euler’s second Proposition). Suppose x divides a2 + b2 and that the quotient, factored into its prime factors is p1 p2 · · · pn . Then a2 + b2 = xp1 p2 · · · pn . If all factors pi can be written as sums of two squares, then we can divide a2 + b2 successively by p1 , p2 , etc., and applying the previous step we deduce that each quotient is a sum of two squares. This until we get to x , concluding that x would have to be the sum of two squares. So, by contraposition, if x is not the sum of two squares, then at least one of the primes pi is not the sum of two squares. 4. If a and b are relatively prime then every factor of a2 + b2 is a sum of two squares. (This is Euler’s Proposition 4. The proof sketched below includes the proof of his Proposition 3). This is the step that uses infinite descent. Let x be a factor of a2 + b2 . We can write a = mx ± c,
b = nx ± d
where c and d are at most half of x in absolute value. This gives: a2 + b2 = m2 x2 ± 2mxc + c2 + n2 x2 ± 2nxd + d2 = Ax + (c2 + d2 ). Therefore, c2 + d2 must be divisible by x , say c2 + d2 = yx . If c and d are not relatively prime, then their gcd must be relatively prime to x (else the common factor of their gcd and x would also be a common factor of a and b which we assume are relatively prime). Thus the square of the gcd divides y (as it divides c2 + d2 ), giving us an expression of the form e2 + f 2 = zx for relatively prime e and f , and with z no more than half of x , since
zx = e2 + f 2 ≤ c2 + d2 ≤
( x )2 2
+
( x )2 2
=
1 2 x . 2
If c and d are relatively prime, then we can use them directly instead of switching to e and f .
174.2. LAGRANGE’S PROOF THROUGH QUADRATIC FORMS
525
If x is not the sum of two squares, then by the third step there must be a factor of z which is not the sum of two squares; call it w . This gives an infinite descent, going from x to a smaller number w , both not the sums of two squares but dividing a sum of two squares. Since an infinite descent is impossible, we conclude that x must be expressible as a sum of two squares, as claimed. 5. Every prime of the form 4n + 1 is a sum of two squares. (This is the main result of Euler’s second paper). If p = 4n+1 , then by Fermat’s Little Theorem each of the numbers 1, 24n , 34n , . . . , (4n)4n is congruent to one modulo p . The differences 24n − 1, 34n − 24n , . . . , (4n)4n − (4n − 1)4n are therefore all divisible by p . Each of these differences can be factored as ( )( ) a4n − b4n = a2n + b2n a2n − b2n . Since p is prime, it must divide one of the two factors. If in any of the 4n − 1 cases it divides the first factor, then by the previous step we conclude that p is itself a sum of two squares (since a and b differ by 1 , they are relatively prime). So it is enough to show that p cannot always divide the second factor. If it divides all 4n − 1 differences 22n − 1, 32n − 22n , . . . , (4n)2n − (4n − 1)2n , then it would divide all 4n − 2 differences of successive terms, all 4n − 3 differences of the differences, and so forth. Since the k th differences of the sequence 1k , 2k , 3k , . . . are all equal to k! (Finite difference), the 2n th differences would all be constant and equal to (2n)! , which is certainly not divisible by p . Therefore, p cannot divide all the second factors which proves that p is indeed the sum of two squares.
174.2 Lagrange’s proof through quadratic forms Lagrange completed a proof in 1775[5] based on his general theory of integral quadratic forms. The following is a slight simplification of his argument, due to Gauss, which appears in article 182 of the Disquisitiones Arithmeticae. A (binary) quadratic form will be taken to be an expression of the form ax2 + 2bxy + cy 2 with a, b, c integers. A number n is said to be represented by the form if there exist integers x, y such that n = ax2 + 2bxy + cy 2 . Fermat’s theorem on sums of two squares is then equivalent to the statement that a prime p is represented by the form x2 + y 2 (i.e., a = c = 1 , b = 0 ) exactly when p is congruent to 1 modulo 4 . The discriminant of the quadratic form is defined to be b2 − ac (this is the definition due to Gauss; Lagrange did not require the xy term to have even coefficient, and defined the discriminant as b2 − 4ac ). The discriminant of x2 + y 2 is then equal to −1 . Two forms ax2 + 2bxy + cy 2 and rx′2 + 2sx′ y ′ + ty ′2 are equivalent if and only if there exist substitutions with integer coefficients
x = αx′ + βy ′ y = γx′ + δy ′ with αδ −βγ = ±1 such that, when substituted into the first form, yield the second. Equivalent forms are readily seen to have the same discriminant. Moreover, it is clear that equivalent forms will represent exactly the same integers. Lagrange proved that all positive definite forms of discriminant −1 are equivalent. Thus, to prove Fermat’s theorem it is enough to find any positive definite form of discriminant −1 that represents p . To do this, it suffices to find an integer m such that p divides m2 + 1 . For, finding such an integer, we can consider the form ( px2 + 2mxy +
m2 + 1 p
) y2 ,
which has discriminant −1 and represents p by setting x = 1 and y = 0. Suppose then that p = 4n + 1. Again we invoke Fermat’s Little Theorem: for any z relatively prime to p, we know that p divides z p−1 − 1 = z 4n − 1 = (z 2n − 1)(z 2n + 1) . Moreover, by a theorem of Lagrange, the number of
526
CHAPTER 174. PROOFS OF FERMAT’S THEOREM ON SUMS OF TWO SQUARES
solutions modulo p to a congruence of degree q modulo p is at most q (this follows since the integers modulo p form a field, and a polynomial of degree q has at most q roots). So the congruence z 2n − 1 ≡ 0 (mod p) has at most 2n solutions among the numbers 1, 2, …, p − 1 = 4n. Therefore, there exists some positive integer z strictly smaller than p (and so relatively prime to p) such that p does not divide z 2n − 1 . Since p divides z 4n − 1 = (z 2n − 1)(z 2n + 1) , p must divide z 2n + 1 . Setting m = z n completes the proof.
174.3 Dedekind’s two proofs using Gaussian integers Richard Dedekind gave at least two proofs of Fermat’s theorem on sums of two squares, both using the arithmetical properties of the Gaussian integers, which are numbers of the form a + bi, where a and b are integers, and i is the square root of −1. One appears in section 27 of his exposition of ideals published in 1877; the second appeared in Supplement XI to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie, and was published in 1894. p−1
1. First proof. If p is an odd prime number, then we have ip−1 = (−1) 2 in the Gaussian integers. Consequently, writing a Gaussian integer ω = x + iy with x,y ∈ Z and applying the Frobenius automorphism in Z[i]/(p), one finds
ω p = (x + yi)p ≡ xp + y p ip ≡ x + (−1)
p−1 2
yi
(mod p),
since the automorphism fixes the elements of Z/(p). In the current case, p = 4n + 1 for some integer n, and so in the above expression for ωp , the exponent (p-1)/2 of −1 is even. Hence the right hand side equals ω, so in this case the Frobenius endomorphism of Z[i]/(p) is the identity. Kummer had already established that if f ∈ {1,2} is the order of the Frobenius automorphism of Z[i]/(p), then the ideal (p) in Z[i] would be a product of 2/f distinct prime ideals. (In fact, Kummer had established a much more general result for any extension of Z obtained by adjoining a primitive m-th root of unity, where m was any positive integer; this is the case m = 4 of that result.) Therefore the ideal (p) is the product of two different prime ideals in Z[i]. Since the Gaussian integers are a Euclidean domain for the norm function N (x + iy) = x2 + y 2 , every ideal is principal and generated by a nonzero element of the ideal of minimal norm. Since the norm is multiplicative, the norm of a generator α of one of the ideal factors of (p) must be a strict divisor of N (p) = p2 , so that we must have p = N (α) = N (a + bi) = a2 + b2 , which gives Fermat’s theorem. 2. Second proof. This proof builds on Lagrange’s result that if p = 4n + 1 is a prime number, then there must be an integer m such that m2 + 1 is divisible by p (we can also see this by Euler’s criterion); it also uses the fact that the Gaussian integers are a unique factorization domain (because they are a Euclidean domain). Since p ∈ Z does not divide either of the Gaussian integers m + i and m − i (as it does not divide their imaginary parts), but it does divide their product m2 + 1 , it follows that p cannot be a prime element in the Gaussian integers. We must therefore have a nontrivial factorization of p in the Gaussian integers, which in view of the norm can have only two factors (since the norm is multiplicative, and p2 = N (p) , there can only be up to two factors of p), so it must be of the form p = (x + yi)(x − yi) for some integers x and y . This immediately yields that p = x2 + y 2 .
174.4 Zagier’s “one-sentence proof” If p = 4k + 1 is prime, then the set S = {(x, y, z) ∈ N3 : x2 + 4yz = p} (here the set N of all natural numbers can be taken to include 0 or to exclude 0, and in both cases, x, y and z must be positive for any (x, y, z) ∈ S, as p is an odd prime) is finite and has two involutions: an obvious one (x, y, z) → (x, z, y), whose fixed points correspond to representations of p as a sum of two squares, and a more complicated one, (x + 2z, z, y − x − z), (x, y, z) 7→ (2y − x, y, x − y + z), (x − 2y, x − y + z, y),
if x < y − z if y − z < x < 2y if x > 2y
which has exactly one fixed point, (1, 1, k); however, the number of fixed points of an involution of a finite set S has the same parity as the cardinality of S, so this number is odd (hence, not zero) for the first involution as well, proving that p is a sum of two squares. This proof, due to Zagier, is a simplification of an earlier proof by Heath-Brown, which in turn was inspired by a proof of Liouville. The technique of the proof is a combinatorial analogue of the topological principle that the
174.5. REFERENCES
527
Euler characteristics of a topological space with an involution and of its fixed point set have the same parity and is reminiscent of the use of sign-reversing involutions in the proofs of combinatorial bijections.
174.5 References • Richard Dedekind, The theory of algebraic integers. • Harold M. Edwards, Fermat’s Last Theorem. A genetic introduction to algebraic number theory. Graduate Texts in Mathematics no. 50, Springer-Verlag, NY, 1977. • C. F. Gauss, Disquisitiones Arithmeticae (English Edition). Transl. by Arthur A. Clarke. Springer-Verlag, 1986. • Goldman, Jay R. (1998), The Queen of Mathematics: A historically motivated guide to Number Theory, A K Peters, ISBN 1-56881-006-7 • D. R. Heath-Brown, Fermat’s two squares theorem. Invariant, 11 (1984) pp. 3–5. • John Stillwell, Introduction to Theory of Algebraic Integers by Richard Dedekind. Cambridge Mathematical Library, Cambridge University Press, 1996. • Don Zagier, A one-sentence proof that every prime p ≡ 1 mod 4 is a sum of two squares. Amer. Math. Monthly 97 (1990), no. 2, 144, doi:10.2307/2323918
174.6 Notes [1] See Goldman’s book, §22.5 [2] Euler à Goldbach, lettre CXXV [3] De numerus qui sunt aggregata quorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 4 (1752/3), 1758, 3-40) [4] Demonstratio theorematis FERMATIANI omnem numerum primum formae 4n+1 esse summam duorum quadratorum. (Novi commentarii academiae scientiarum Petropolitanae 5 (1754/5), 1760, 3-13) [5] Nouv. Mém. Acad. Berlin, année 1771, 125; ibid. année 1773, 275; ibid année 1775, 351.
174.7 External links • Two more proofs at PlanetMath.org • “A one-sentence proof of the theorem”. Archived from the original on 5 February 2012. • reprint of Heath-Brown’s proof, with commentary
Chapter 175
Proofs of quadratic reciprocity In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusual number of proofs. Several hundred proofs of the law of quadratic reciprocity have been found.
175.1 Proofs that are accessible Of relatively elementary, combinatorial proofs, there are two which apply types of double counting. One by Gotthold Eisenstein counts lattice points. Another applies Zolotarev’s lemma to Z/pqZ expressed by the Chinese remainder theorem as Z/pZ×Z/qZ, and calculates the signature of a permutation.
175.2 Eisenstein’s proof Eisenstein’s proof of quadratic reciprocity is a simplification of Gauss’s third proof. It is more geometrically intuitive and requires less technical manipulation. The point of departure is “Eisenstein’s lemma”, which states that for distinct odd primes p, q, ( ) ∑ q = (−1) u ⌊qu/p⌋ , p where ⌊x⌋ denotes the floor function (the largest integer less than or equal to x), and where the sum is taken over the even integers u = 2, 4, 6, ..., p−1. For example, (
7 11
)
= (−1)⌊14/11⌋+⌊28/11⌋+⌊42/11⌋+⌊56/11⌋+⌊70/11⌋ = (−1)1+2+3+5+6 = (−1)17 = −1.
This result is very similar to Gauss’s lemma, and can be proved in a similar fashion (proof given below). Using this representation of (q/p), the main argument is quite elegant. The sum Σu ⌊qu/p⌋ counts the number of lattice points with even x-coordinate in the interior of the triangle ABC in the following diagram: Because each column has an even number of points (namely q−1 points), the number of such lattice points in the region BCYX is the same modulo 2 as the number of such points in the region CZY: Then by flipping the diagram in both axes, we see that the number of points with even x-coordinate inside CZY is the same as the number of points inside AXY having odd x-coordinates: The conclusion is that ( ) q = (−1)µ , p 528
175.2. EISENSTEIN’S PROOF
529
7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 The number of points with even x-coordinate inside BCYX (marked by O’s) is equal modulo 2 to the number of such points in CZY (marked by X’s)
where μ is the total number of lattice points in the interior of AYX. Switching p and q, the same argument shows that ( ) p = (−1)ν , q where ν is the number of lattice points in the interior of WYA. Since there are no lattice points on the line AY itself (because p and q are relatively prime), and since the total number of points in the rectangle WYXA is (
p−1 2
)(
q−1 2
) ,
we obtain finally ( )( ) q p = (−1)µ+ν = (−1)(p−1)(q−1)/4 . p q
175.2.1
Proof of Eisenstein’s lemma
For an even integer u in the range 1 ≤ u ≤ p−1, denote by r(u) the least positive residue of qu modulo p. (For example, for p = 11, q = 7, we allow u = 2, 4, 6, 8, 10, and the corresponding values of r(u) are 3, 6, 9, 1, 4.) The numbers (−1)r(u) r(u), again treated as least positive residues modulo p, are all even (in our running example, they are 8, 6, 2, 10, 4.) Furthermore, they are all distinct, because if (−1)r(u) r(u) ≡ (−1)r(t) r(t) mod p, then we may divide out by q to obtain u ≡ ±t mod p. This forces u ≡ t mod p, because both u and t are even, whereas p is odd. Since there exactly (p−1)/2 of them and they are distinct, they must be simply a rearrangement of the even integers 2, 4, ..., p−1. Multiplying them together, we obtain
530
CHAPTER 175. PROOFS OF QUADRATIC RECIPROCITY
7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 The number of points with even x-coordinate inside CZY is equal to the number of points with odd x-coordinate inside AXY
(−1)r(2) 2q · (−1)r(4) 4q · · · · · (−1)r(p−1) (p − 1)q ≡ 2 · 4 · · · · · (p − 1) (mod p). Dividing out successively by 2, 4, ..., p−1 on both sides (which is permissible since none of them are divisible by p) and rearranging, we have
q (p−1)/2 ≡ (−1)r(2)+r(4)+···+r(p−1) (mod p). On the other hand, by the definition of r(u) and the floor function, ⌊ ⌋ qu r(u) qu = + , p p p and so since p is odd and u is even, we see that ⌊qu/p⌋ and r(u) are congruent modulo 2. Finally this shows that
q (p−1)/2 ≡ (−1)
∑
u ⌊qu/p⌋
(mod p).
We are finished because the left hand side is just an alternative expression for (q/p).
175.3 Proof using algebraic number theory The proof presented here is by no means the simplest known; however, it is quite a deep one, in the sense that it motivates some of the ideas of Artin reciprocity.
175.3. PROOF USING ALGEBRAIC NUMBER THEORY
175.3.1
531
Cyclotomic field setup
Suppose that p is an odd prime. The action takes place inside the cyclotomic field
L = Q(ζp ), where ζ is a primitive pth root of unity. The basic theory of cyclotomic fields informs us that there is a canonical isomorphism G = Gal(L/Q) ∼ = (Z/pZ)× , which sends the automorphism σa satisfying
σa (ζp ) = ζpa to the element
a ∈ (Z/pZ)× . (This is because the morphism of reduction from Z to Z/qZ is injective on the set of p-th roots of unity) Now consider the subgroup H of squares of elements of G. Since G is cyclic, H has index 2 in G, so the subfield corresponding to H under the Galois correspondence must be a quadratic extension of Q. (In fact it is the unique quadratic extension of Q contained in L.) The Gaussian period theory determines which one; it turns out to be √ Q( p∗ ), where { p p = −p ∗
if p = 1 (mod 4), if p = 3 (mod 4).
At this point we start to see a hint of quadratic reciprocity emerging from our framework. On one hand, the image of H in
(Z/pZ)× consists precisely of the (nonzero) quadratic residues modulo p. On the other hand, H is related to an attempt to take the square root of p (or possibly of −p). In other words, if now q is an odd prime (different from p), we have so far shown that ( ) q =1 p
175.3.2
⇐⇒
σq ∈ H
⇐⇒
√ σq fixes Q( p∗ ).
The Frobenius automorphism
Choose any prime ideal β of the ring of integers OL lying over q, which is unramified, and let
ϕ ∈ Gal(L/Q)
532
CHAPTER 175. PROOFS OF QUADRATIC RECIPROCITY
be the Frobenius automorphism associated to β; the characteristic property of ϕ is that
ϕ(x) ≡ xq (mod β) for any x in OL. (The existence of such a Frobenius element depends on quite a bit of algebraic number theory machinery.) The key fact about ϕ that we need is that for any subfield K of L,
⇐⇒
ϕ fixes K
q splits completely in K.
Indeed, let δ be any ideal of OK below β (and hence above q). Then, since
ϕ(x) ≡ xq (mod δ) for any x in OK, we see that
ϕ|K ∈ Gal(K/Q) is a Frobenius for δ. A standard result concerning ϕ is that its order is equal to the corresponding inertial degree; that is,
ord(ϕ|K ) = [OK /δOK : Z/qZ]. The left hand side is equal to 1 if and only if φ fixes K, and the right hand side is equal to one if and only q splits completely in K, so we are done. Now, since the pth roots of unity are distinct modulo β (i.e. the polynomial Xp − 1 is separable in characteristic q), we must have
ϕ(ζp ) = ζpq ; that is, ϕ coincides with the automorphism σq defined earlier. Taking K to be the quadratic field in which we are interested, we obtain the equivalence ( ) q =1 p
175.3.3
⇐⇒
√ q splits completely in Q( p∗ ).
Completing the proof
Finally we must show that √ q splits completely in Q( p∗ )
( ⇐⇒
p∗ q
) = 1.
Once we have done this, the law of quadratic reciprocity falls out immediately since (
p∗ q
) =
( ) q p
if p = 1 mod 4, and
175.3. PROOF USING ALGEBRAIC NUMBER THEORY
(
p∗ q
)
( =
−p q
)
( =
−1 q
( ) )( ) + pq p ( ) = − p q q
if q = 1 (mod 4), if q = 3 (mod 4)
if p = 3 mod 4. To show the last equivalence, suppose first that (
p∗ q
) = 1.
In this case, there is some integer x (not divisible by q) such that
x2 ≡ p∗ (mod q), say
x2 − p∗ = cq for some integer c. Let √ K = Q( p∗ ), and consider the ideal
(x −
√
p∗ , q)
of K. It certainly divides the principal ideal (q). It cannot be equal to (q), since
x−
√ ∗ p
is not divisible by q. It cannot be the unit ideal, because then
(x +
√
p∗ ) = (x +
√ ∗ √ √ p )(x − p∗ , q) = (cq, q(x + p∗ ))
is divisible by q, which is again impossible. Therefore (q) must split in K. Conversely, suppose that (q) splits, and let β be a prime of K above q. Then
(q) ⊊ β, so we may choose some √ a + b p∗ ∈ β \ (q), where a and b are in Q. Actually, since
p∗ = 1 (mod 4),
533
534
CHAPTER 175. PROOFS OF QUADRATIC RECIPROCITY
elementary theory of quadratic fields implies that the ring of integers of K is precisely [ Z
1+
√ ∗] p , 2
so the denominators of a and b are at worst equal to 2. Since q ≠ 2, we may safely multiply a and b by 2, and assume that √ a + b p∗ ∈ β \ (q), where now a and b are in Z. In this case we have √ √ (a + b p∗ )(a − b p∗ ) = a2 − b2 p∗ ∈ β ∩ Z = (q), so q | a2 − b2 p∗ . However, q cannot divide b, since then also q divides a, which contradicts our choice of √ a + b p∗ . Therefore, we may divide by b modulo q, to obtain p∗ = (ab−1 )2 (mod q) as desired.
175.4 References Every textbook on elementary number theory (and quite a few on algebraic number theory) has a proof of quadratic reciprocity. Two are especially noteworthy: Franz Lemmermeyer’s Reciprocity Laws: From Euler to Eisenstein has many proofs (some in exercises) of both quadratic and higher-power reciprocity laws and a discussion of their history. Its immense bibliography includes literature citations for 196 different published proofs. Kenneth Ireland and Michael Rosen’s A Classical Introduction to Modern Number Theory also has many proofs of quadratic reciprocity (and many exercises), and covers the cubic and biquadratic cases as well. Exercise 13.26 (p 202) says it all Count the number of proofs to the law of quadratic reciprocity given thus far in this book and devise another one. • Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-54066957-4 • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (second edition), New York: Springer, ISBN 0-387-97329-X • G. Rousseau. “On the Quadratic Reciprocity Law”, J. Austral. Math. Soc. (Series A), v51, 1991, 423–425. (online) • L. Washington. Introduction to Cyclotomic Fields, 2nd ed.
175.5. EXTERNAL LINKS
175.5 External links • Chronology of Proofs of the Quadratic Reciprocity Law (233 proofs!)
535
Chapter 176
Pseudo algebraically closed field In mathematics, a field K is pseudo algebraically closed if it satisfies certain properties which hold for any algebraically closed field. The concept was introduced by James Ax in 1967.[1]
176.1 Formulation A field K is pseudo algebraically closed (usually abbreviated by PAC[2] ) if one of the following equivalent conditions holds: • Each absolutely irreducible variety V defined over K has a K -rational point. ∂f • For each absolutely irreducible polynomial f ∈ K[T1 , T2 , · · · , Tr , X] with ∂X ̸= 0 and for each nonzero r+1 g ∈ K[T1 , T2 , · · · , Tr ] there exists (a, b) ∈ K such that f (a, b) = 0 and g(a) ̸= 0 .
• Each absolutely irreducible polynomial f ∈ K[T, X] has infinitely many K -rational points. • If R is a finitely generated integral domain over K with quotient field which is regular over K , then there exist a homomorphism h : R → K such that h(a) = a for each a ∈ K
176.2 Examples • Algebraically closed fields and separably closed fields are always PAC. • Pseudo-finite fields and hyper-finite fields are PAC. • A non-principal ultraproduct of distinct finite fields is (pseudo-finite and hence[3] ) PAC.[2] Ax deduces this from the Riemann hypothesis for curves over finite fields.[1] • Infinite algebraic extensions of finite fields are PAC.[4] • The PAC Nullstellensatz. The absolute Galois group G of a field K is profinite, hence compact, and hence equipped with a normalized Haar measure. Let K be a countable Hilbertian field and let e be a positive integer. Then for almost all e -tuple (σ1 , ..., σe ) ∈ Ge , the fixed field of the subgroup generated by the automorphisms is PAC. Here the phrase “almost all” means “all but a set of measure zero”.[5] (This result is a consequence of Hilbert’s irreducibility theorem.) • Let K be the maximal totally real Galois extension of the rational numbers and i the square root of −1. Then K(i) is PAC. 536
176.3. PROPERTIES
537
176.3 Properties • The Brauer group of a PAC field is trivial,[6] as any Severi–Brauer variety has a rational point.[7] • The absolute Galois group of a PAC field is a projective profinite group; equivalently, it has cohomological dimension at most 1.[7] • A PAC field of characteristic zero is C1.[8]
176.4 References [1] Fried & Jarden (2008) p.218 [2] Fried & Jarden (2008) p.192 [3] Fried & Jarden (2008) p.449 [4] Fried & Jarden (2008) p.196 [5] Fried & Jarden (2008) p.380 [6] Fried & Jarden (2008) p.209 [7] Fried & Jarden (2008) p.210 [8] Fried & Jarden (2008) p.462
• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.
Chapter 177
Pseudo-finite field In mathematics, a pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over F has a point defined over F). Every hyperfinite field is pseudo-finite and every pseudo-finite field is quasifinite. Every non-principal ultraproduct of finite fields is pseudofinite. Pseudo-finite fields were introduced by Ax (1968).
177.1 References • Ax, James (1968), “The Elementary Theory of Finite Fields”, Annals of Mathematics, Second Series (Annals of Mathematics) 88 (2): 239–271, doi:10.2307/1970573, ISSN 0003-486X, MR 0229613, Zbl 0195.05701 • Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.), Springer-Verlag, pp. 448–453, ISBN 978-3-540-77269-9, Zbl 1145.12001
538
Chapter 178
Purely inseparable extension In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form xq = a, with q a power of p and a in k. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions.
178.1 Purely inseparable extensions An algebraic extension E ⊇ F is a purely inseparable extension if and only if for every α ∈ E \ F , the minimal polynomial of α over F is not a separable polynomial.[1] If F is any field, the trivial extension F ⊇ F is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section. Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known. If E ⊇ F is an algebraic extension with (non-zero) prime characteristic p, then the following are equivalent:[2] 1. E is purely inseparable over F. n
2. For each element α ∈ E , there exists n ≥ 0 such that αp ∈ F . n
3. Each element of E has minimal polynomial over F of the form X p − a for some integer n ≥ 0 and some element a∈F . It follows from the above equivalent characterizations that if E = F [α] (for F a field of prime characteristic) such n that αp ∈ F for some integer n ≥ 0 , then E is purely inseparable over F.[3] (To see this, note that the set of all x n such that xp ∈ F for some n ≥ 0 forms a field; since this field contains both α and F, it must be E, and by condition 2 above, E ⊇ F must be purely inseparable.) If F is an imperfect field of prime characteristic p, choose a ∈ F such that a is not a pth power in F, and let f(X) = Xp − a. Then f has no root in F, and so if E is a splitting field for f over F, it is possible to choose α with f (α) = 0 . In particular, αp = a and by the property stated in the paragraph directly above, it follows that F [α] ⊇ F is a non-trivial purely inseparable extension (in fact, E = F [α] , and so E ⊇ F is automatically a purely inseparable extension).[4] Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic. If K is a field of characteristic p, and if V is an algebraic variety over K of dimension greater than zero, the function field K(V) is a purely inseparable extension over the subfield K(V)p of pth powers (this follows from condition 2 above). Such extensions occur in the context of multiplication by p on an elliptic curve over a finite field of characteristic p.
178.1.1
Properties
• If the characteristic of a field F is a (non-zero) prime number p, and if E ⊇ F is a purely inseparable extension, then if F ⊆ K ⊆ E , K is purely inseparable over F and E is purely inseparable over K. Furthermore, if [E : F] is finite, then it is a power of p, the characteristic of F.[5] 539
540
CHAPTER 178. PURELY INSEPARABLE EXTENSION
• Conversely, if F ⊆ K ⊆ E is such that F ⊆ K and K ⊆ E are purely inseparable extensions, then E is purely inseparable over F.[6] • An algebraic extension E ⊇ F is an inseparable extension if and only if there is some α ∈ E \ F such that the minimal polynomial of α over F is not a separable polynomial (i.e., an algebraic extension is inseparable if and only if it is not separable; note, however, that an inseparable extension is not the same thing as a purely inseparable extension). If E ⊇ F is a finite degree non-trivial inseparable extension, then [E : F] is necessarily divisible by the characteristic of F.[7] • If E ⊇ F is a finite degree normal extension, and if K = Fix(Gal(E/F )) , then K is purely inseparable over F and E is separable over K.[8]
178.2 Galois correspondence for purely inseparable extensions Jacobson (1937, 1944) introduced a variation of Galois theory for purely inseparable extensions of exponent 1, where the Galois groups of field automorphisms in Galois theory are replaced by restricted Lie algebras of derivations. The simplest case is for finite index purely inseparable extensions K⊆L of exponent at most 1 (meaning that the pth power of every element of L is in K). In this case the Lie algebra of K-derivations of L is a restricted Lie algebra that is also a vector space of dimension n over L, where [L:K] = pn , and the intermediate fields in L containing K correspond to the restricted Lie subalgebras of this Lie algebra that are vector spaces over L. Although the Lie algebra of derivations is a vector space over L, it is not in general a Lie algebra over L, but is a Lie algebra over K of dimension n[L:K] = npn . A purely separable extension is called a modular extension if it is a tensor product of simple extensions, so in particular every extension of exponent 1 is modular, but there are non-modular extensions of exponent 2 (Weisfeld 1965). Sweedler (1968) and Gerstenhaber & Zaromp (1970) gave an extension of the Galois correspondence to modular purely inseparable extensions, where derivations are replaced by higher derivations.
178.3 See also • Jacobson–Bourbaki theorem
178.4 References [1] Isaacs, p. 298 [2] Isaacs, Theorem 19.10, p. 298 [3] Isaacs, Corollary 19.11, p. 298 [4] Isaacs, p. 299 [5] Isaacs, Corollary 19.12, p. 299 [6] Isaacs, Corollary 19.13, p. 300 [7] Isaacs, Corollary 19.16, p. 301 [8] Isaacs, Theorem 19.18, p. 301
• Gerstenhaber, Murray; Zaromp, Avigdor (1970), “On the Galois theory of purely inseparable field extensions”, Bulletin of the American Mathematical Society 76: 1011–1014, doi:10.1090/S0002-9904-1970-12535-6, ISSN 0002-9904, MR 0266904 • Isaacs, I. Martin (1993), Algebra, a graduate course (1st ed.), Brooks/Cole Publishing Company, ISBN 0-53419002-2 • Jacobson, Nathan (1937), “Abstract Derivation and Lie Algebras”, Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 42 (2): 206–224, doi:10.2307/1989656, ISSN 0002-9947
178.4. REFERENCES
541
• Jacobson, Nathan (1944), “Galois theory of purely inseparable fields of exponent one”, American Journal of Mathematics 66: 645–648, doi:10.2307/2371772, ISSN 0002-9327, MR R0011079 • Sweedler, Moss Eisenberg (1968), “Structure of inseparable extensions”, Annals of Mathematics. Second Series 87: 401–410, doi:10.2307/1970711, ISSN 0003-486X, MR 0223343[http://www.jstor.org/stable/1970818 correction] • Weisfeld, Morris (1965), “Purely inseparable extensions and higher derivations”, Transactions of the American Mathematical Society 116: 435–449, doi:10.2307/1994126, ISSN 0002-9947, MR 0191895
Chapter 179
Pythagoras number Not to be confused with Pythagoras’s constant. In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number p(K) of a field K is the smallest positive integer p such that every sum of squares in K is a sum of p squares. A Pythagorean field is one with Pythagoras number 1: that is, every sum of squares is already a square.
179.1 Examples • Every positive real is a square, so p(R) = 1. • For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares,[1] so p = 2. • By Lagrange’s four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so p(Q) = 4.
179.2 Properties • Every positive integer occurs as the Pythagoras number of some formally real field.[2] • The Pythagoras number is related to the Stufe by p(F) ≤ s(F) + 1.[3] If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1,[4] and both cases are possible: for F = C we have s = p = 1, whereas for F = F5 we have s = 1, p = 2.[5] • The Pythagoras number is related to the height of a field F: if F is formally real then h(F) is the smallest power of 2 which is not less than p(F); if F is not formally real then h(F) = 2s(F).[6] As a consequence, the Pythagoras number of a non formally real field, if finite, is either a power of 2 or 1 less than a power of 2, and all cases occur.[7]
179.3 Notes [1] Lam (2005) p. 36 [2] Lam (2005) p. 398 [3] Rajwade (1993) p. 44 [4] Rajwade (1993) p. 228 [5] Rajwade (1993) p. 261
542
179.4. REFERENCES
543
[6] Lam (2005) p. 395 [7] Lam (2005) p. 396
179.4 References • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
Chapter 180
Pythagorean field In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element √1 + λ2 for some λ in F. So a Pythagorean field is one closed under taking Pythagorean extensions. For any field F there is a minimal Pythagorean field F py containing it, unique up to isomorphism, called its Pythagorean closure.[1] The Hilbert field is the minimal ordered Pythagorean field.[2]
180.1 Properties Every Euclidean field (an ordered field in which all positive elements are squares) is an ordered Pythagorean field, but the converse does not hold.[3] A quadratically closed field is Pythagorean field but not conversely (R is Pythagorean); however, a non formally real Pythagorean field is quadratically closed.[4] The Witt ring of a Pythagorean field is of order 2 if the field is not formally real, and torsion-free otherwise.[1] For a field F there is an exact sequence involving the Witt rings
0 → TorIW (F ) → W (F ) → W (F py ) where I W(F) is the fundamental ideal of the Witt ring of F [5] and Tor I W(F) denotes its torsion subgroup (which is just the nilradical of W(F).[6]
180.1.1
Equivalent conditions
The following conditions on a field F are equivalent to F being Pythagorean: • The general u-invariant u(F) is 0 or 1.[7] • If ab is not a square in F then there is an order on F for which a, b have different signs.[8] • F is the intersection of its Euclidean closures.[9]
180.2 Models of geometry Pythagorean fields can be used to construct models for some of Hilbert’s axioms for geometry (Ito 1980, 163 C). The coordinate geometry given by F n for F a Pythagorean field satisfies many of Hilbert’s axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels. However, in general this geometry need not satisfy all Hilbert’s axioms unless the field F has extra properties: for example, if the field is also ordered then the geometry will satisfy Hilbert’s ordering axioms, and if the field is also complete the geometry will satisfy Hilbert’s completeness axiom. 544
180.3. DILLER–DRESS THEOREM
545
The Pythagorean closure of a non-archimedean ordered field, such as the Pythagorean closure of the field of rational functions Q(t) in one variable over the rational numbers Q, can be used to construct non-archimedean geometries that satisfy many of Hilbert’s axioms but not his axiom of completeness.[10] Dehn used such a field to construct two Dehn planes, examples of non-Legendrian geometry and semi-Euclidean geometry respectively, in which there are many lines though a point not intersecting a given line but where the sum of the angles of a triangle is at least π.[11]
180.3 Diller–Dress theorem This theorem states that if E/F is a finite field extension, and E is Pythagorean, then so is F.[12] As a consequence, no algebraic number field is Pythagorean, since all such fields are finite over Q, which is not Pythagorean.[13]
180.4 Superpythagorean fields A superpythagorean field F is a formally real field with the property that if S is a subgroup of index 2 in F ∗ and does not contain −1, then S defines a ordering on F. An equivalent definition is that F is a formally real field in which the set of squares forms a fan. A superpythagorean field is necessarily Pythagorean.[12] The analogue of the Diller–Dress theorem holds: if E/F is a finite extension and E is superpythagorean then so is F.[14] In the opposite direction, if F is superpythagorean and E is a formally real field containing F and contained in the quadratic closure of F then E is superpythagorean.[15]
180.5 Notes [1] Milnor & Husemoller (1973) p. 71 [2] Greenberg (2010) [3] Martin (1998) p. 89 [4] Rajwade (1993) p.230 [5] Milnor & Husemoller (1973) p. 66 [6] Milnor & Husemoller (1973) p. 72 [7] Lam (2005) p.410 [8] Lam (2005) p.293 [9] Efrat (2005) p.178 [10] (Ito 1980, 163 D) [11] Dehn (1900) [12] Lam (1983) p.45 [13] Lam (2005) p.269 [14] Lam (1983) p.47 [15] Lam (1983) p.48
180.6 References • Dehn, Max (1900), “Die Legendre’schen Sätze über die Winkelsumme im Dreieck”, Mathematische Annalen 53 (3): 404–439, doi:10.1007/BF01448980, ISSN 0025-5831, JFM 31.0471.01 • Efrat, Ido (2006), Valuations, orderings, and Milnor K-theory, Mathematical Surveys and Monographs 124, Providence, RI: American Mathematical Society, ISBN 0-8218-4041-X, Zbl 1103.12002
546
CHAPTER 180. PYTHAGOREAN FIELD
• Elman, Richard; Lam, T. Y. (1972), “Quadratic forms over formally real fields and pythagorean fields”, American Journal of Mathematics 94: 1155–1194, ISSN 0002-9327, JSTOR 2373568, MR 0314878 • Greenberg, Marvin J. (2010), “Old and new results in the foundations of elementary plane Euclidean and non-Euclidean geometries”, Am. Math. Mon. 117 (3): 117–219, ISSN 0002-9890, Zbl 1206.51015 • Iyanaga, Shôkichi; Kawada, Yukiyosi, eds. (1980) [1977], Encyclopedic dictionary of mathematics, Volumes I, II, Translated from the 2nd Japanese edition, paperback version of the 1977 edition (1st ed.), MIT Press, ISBN 978-0-262-59010-5, MR 591028 • Lam, T. Y. (1983), Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics 52, American Mathematical Society, ISBN 0-8218-0702-1, Zbl 0516.12001 • Lam, T. Y. (2005), “Chapter VIII section 4: Pythagorean fields”, Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67, Providence, R.I.: American Mathematical Society, pp. 255–264, ISBN 978-0-8218-1095-8, MR 2104929 • Martin, George E. (1998), Geometric Constructions, Undergraduate Texts in Mathematics, Springer-Verlag, ISBN 0-387-98276-0 • Milnor, J.; Husemoller, D. (1973), Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag, ISBN 3-540-06009-X, Zbl 0292.10016 • Rajwade, A. R. (1993), Squares, London Mathematical Society Lecture Note Series 171, Cambridge University Press, ISBN 0-521-42668-5, Zbl 0785.11022
Chapter 181
Quadratic field In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers. The map d ↦ Q(√d) is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields. If d > 0 the corresponding quadratic field is called a real quadratic field, and for d < 0 an imaginary quadratic field or complex quadratic field, corresponding to whether it is or not a subfield of the field of the real numbers. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important.
181.1 Ring of integers Main article: Quadratic integer
181.2 Discriminant For a nonzero square free integer d, the discriminant of the quadratic field K=Q(√d) is d if d is congruent to 1 modulo 4, and otherwise 4d. For example, when d is −1 so that K is the field of so-called Gaussian rationals, the discriminant is −4. The reason for this distinction relates to general algebraic number theory. The ring of integers of K is spanned over the rational integers by 1 and √d only in the second case, while in the first case it is spanned by 1 and (1 + √d)/2. The set of discriminants of quadratic fields is exactly the set of fundamental discriminants.
181.3 Prime factorization into ideals Any prime number p gives rise to an ideal pOK in the ring of integers OK of a quadratic field K. In line with general theory of splitting of prime ideals in Galois extensions, this may be p is inert (p) is a prime ideal The quotient ring is the finite field with p2 elements: OK/pOK = Fp2 p splits (p) is a product of two distinct prime ideals of OK. The quotient ring is the product OK/pOK = Fp × Fp. p is ramified (p) is the square of a prime ideal of OK. The quotient ring contains non-zero nilpotent elements. 547
548
CHAPTER 181. QUADRATIC FIELD
The third case happens if and only if p divides the discriminant D. The first and second cases occur when the Kronecker symbol (D/p) equals −1 and +1, respectively. For example, if p is an odd prime not dividing D, then p splits if and only if D is congruent to a square modulo p. The first two cases are in a certain sense equally likely to occur as p runs through the primes, see Chebotarev density theorem.[1] The law of quadratic reciprocity implies that the splitting behaviour of a prime p in a quadratic field depends only on p modulo D, where D is the field discriminant.
181.4 Quadratic subfields of cyclotomic fields 181.4.1
The quadratic subfield of the prime cyclotomic field
A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive p-th root of unity, with p a prime number > 2. The uniqueness is a consequence of Galois theory, there being a unique subgroup of index 2 in the Galois group over Q. As explained at Gaussian period, the discriminant of the quadratic field is p for p = 4n + 1 and −p for p = 4n + 3. This can also be predicted from enough ramification theory. In fact p is the only prime that ramifies in the cyclotomic field, so that p is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants −4p and 4p in the respective cases.
181.4.2
Other cyclotomic fields
If one takes the other cyclotomic fields, they have Galois groups with extra 2-torsion, and so contain at least three quadratic fields. In general a quadratic field of field discriminant D can be obtained as a subfield of a cyclotomic field of D-th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the Führerdiskriminantenproduktformel.
181.5 Orders of quadratic number fields of small discriminant The following table shows some orders of small discriminant of quadratic fields, together with some degenerate cases when the discriminant is a square and the corresponding quadratic extension of Z is not an integral domain.
181.6 See also • Eisenstein–Kronecker number • Heegner number • Infrastructure (number theory) • Quadratic integer • Quadratic irrational • Stark–Heegner theorem
181.7 Notes [1] Samuel, pp. 76–77
181.8. REFERENCES
549
181.8 References • Buell, Duncan (1989). Binary quadratic forms: classical theory and modern computations. Springer-Verlag. ISBN 0-387-97037-1. Chapter 6. • Samuel, Pierre (1972). Algebraic number theory. Hermann/Kershaw. • Stewart, I. N.; Tall, D. O. (1979). Algebraic number theory. Chapman and Hall. ISBN 0-412-13840-9. Chapter 3.1.
181.9 External links • Weisstein, Eric W., “Quadratic Field”, MathWorld. • Hazewinkel, Michiel, ed. (2001), “Quadratic field”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4
Chapter 182
Quadratic integer In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are the solutions of equations of the form x2 + Bx + C = 0 with B and C integers. They are thus algebraic integers of the degree two. When algebraic integers are considered, usual integers are often called rational integers. Common examples of quadratic integers are the square roots of integers, such as √2, and the complex number i = √–1, which generates the Gaussian integers. Another common example is the non-real cubic root of unity −1 + √–3/2, which generates the Eisenstein integers. Quadratic integers occur in the solutions of many Diophantine equations, such as Pell’s equations. The study of rings of quadratic integers is basic for many questions of algebraic number theory.
182.1 History Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same D, which allowed them to solve some cases of Pell’s equation. The characterization of the quadratic integers was first given by Richard Dedekind in 1871.[1][2]
182.2 Definition A quadratic integer is a complex number which is a solution of an equation of the form x2 + Bx + C = 0 with B and C integers. In other words, a quadratic integer is an algebraic integer in a quadratic field. Each quadratic integer that is not an integer lies in a uniquely determined quadratic field, √ namely, the extension of Q generated by the square-root of B2 −4C, which can always be written in the form Q( D), where D is the unique square-free integer for which B2 – 4C = DE 2 for some integer E. √ The quadratic integers (including the√ordinary integers), which belong to a quadratic fieds Q( D), form a integral domain called ring of integers of Q( D). Here and in the following, D is supposed to be a square-free √ does not restricts the generality, as the √ integer. This equality √a2 D = a√D (for any positive integer a) implies Q( D) = Q( a2 D). Every quadratic integer may be written a + ωb , where a and b are integers, and where ω is defined by: 550
182.3. NORM AND CONJUGATION
ω=
{√ D √
1+ D 2
551
if D ≡ 2, 3 (mod 4) if D ≡ 1 (mod 4)
(as D has been supposed square-free the case D ≡ 0 (mod 4) is impossible, since it would imply that D would be divisible by the square 4). Although the quadratic integers belonging to a given field form a ring, the set of all quadratic integers is not √ quadratic √ a ring, because it is not closed under addition, as 2 + 3 is an algebraic integer, which has a minimal polynomial of degree four.
182.3 Norm and conjugation √ A quadratic integer in Q( D) may be written a + b√D, where either a and b are either integers, or, only if D ≡ 1 (mod 4), halves of odd integers. The norm of such a quadratic integer is N(a + b√D) = a2 – b2 D. The norm of a quadratic integer is always an integer. If D < 0, the norm of a quadratic integer is the square of its absolute value as a complex number (this is false if D > 0). The norm is a completely multiplicative function, which means that the norm of a product of quadratic integers is always the product of their norms. Every quadratic integer a + b√D has a conjugate √ √ a + b D = a − b D. An algebraic integer has the same norm as its conjugate, and this norm is the product of the algebraic integer and its conjugate. The conjugate of a sum or a product of algebraic integers it the sum or the product (respectively) of the √ conjugates. This means that the conjugation is an automorphism of the ring of the integers of Q( D).
182.4 Units √ A quadratic integer is a unit in the ring of the integers of Q( D) if and only if its norm is 1 or –1. In the first case its multiplicative inverse is its conjugate. It is the opposite of its conjugate in the second case. √ If D < 0, the ring of the integers of Q( D) has at most six units. In the case of the Gaussian integers (D = –1), the four units are 1, –1, √–1, –√–1. In the case of the Eisenstein integers (D = –3), the six units are ±1, ±1 ± √–3/2. For all other negative D, there are only two units that are 1 and –1. √ If D > 0, the ring of the integers of Q( D) has infinitely many units that are equal to ±ui , where i is an arbitrary integer, and u is a particular unit called a fundamental unit. Given a fundamental unit u, there are three other fundamental units, its conjugate u, and also −u and −u. Commonly, one calls the fundamental unit, the unique one which has an absolute value greater than 1 (as a real number). It is the unique fundamental unit that may be written a + b√D, with a and b positive (integers or halves of integers). The fundamental units for the 10 smallest positive square-free D are 1 + √2, 2 + √3, 1 + √5/2 (the golden ratio), 5 + 2√6, 8 + 3√7, 3 + √10, 10 + 3√11, 3 + √13/2, 15 + 4√14, 4 + √15. For larger D, the coefficients of the fundamental unit may be very large. For example, for D = 19, 31, 43, the fundamental units are respectively 170 + 39 √19, 1520 + 273 √31 and 3482 + 531 √43.
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CHAPTER 182. QUADRATIC INTEGER
182.5 Quadratic integer rings Every square-free integer (different of √ 0 and 1) D defines a quadratic integer ring, which is the integral domain of the algebraic integers contained in Q( D). It is the set Z[ω] =a + ωb : a, b ∈ Z, where ω is defined as above. It is called the ring of integers of Q(√D) and often denoted OQ(√D) . By definition, it is the integral closure of Z in √ Q( D). The properties of the quadratic integers (and more generally of algebraic integers) has been a long standing problem, which has motivated the elaboration of the notions of ring and ideal. In particular the fundamental theorem of arithmetic is not true in many rings of quadratic integers. However there is a unique factorization for ideals, which is expressed by the fact that every ring of algebraic integers is a Dedekind domain. Quadratic integer rings and their associated quadratic fields are thus commonly the starting examples of most studies of algebraic number fields. The quadratic integer rings divide in two classes depending on the sign of D. If D > 0, all elements of OQ(√D) are real, and the ring is a real quadratic integer ring. If D < 0, the only real elements of OQ(√D) are the ordinary integers, and the ring is a complex quadratic integer ring.
182.5.1
Examples of complex quadratic integer rings
For D < 0, ω is a complex (imaginary or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic complex numbers. √ • A classic example is Z[ −1] , the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.[3] [ √ ] • The elements in OQ(√−3) = Z 1+ 2 −3 are called Eisenstein integers. Both rings mentioned above are rings of integers of cyclotomic fields Q(ζ4 ) and Q(ζ3 ) correspondingly. In contrast, Z[√−3] is not even a Dedekind domain. Both above examples are principal ideal rings and also Euclidean domains for the norm. This is not the case for
OQ(√−5) = Z
[√ ] −5 ,
which is not even a unique factorization domain. This can be shown as follows. In OQ(√−5) , we have √ √ −5)(2 − −5). √ √ The factors 3, 2 + −5 and 2 − −5 are irreducible, as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of ±1 being at least 4. Thus the factorization of 9 into irreducible factors is not unique. √ √ The ideals ⟨3, 1 + −5⟩ and ⟨3, 1 − −5⟩ are not principal, as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible. 9 = 3 . 3 = (2 +
182.5.2
Examples of real quadratic integer rings
For D > 0, ω is a positive irrational real number, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the Pell’s equation X2 − D Y 2 = 1, a Diophantine equation that has been widely studied, are the units of these rings, for D ≡ 2, 3 (mod 4). • For D = 5, ω = 1+√5/2 is the golden ratio. This ring was studied by Peter Gustav Lejeune Dirichlet. Its invertible elements have the form ±ωn , where n is an arbitrary integer. This ring also arises from studying 5-fold rotational symmetry on Euclidean plane, for example, Penrose tilings.[4]
182.5. QUADRATIC INTEGER RINGS
553
i 9 8 7 6 5 4 3 2 1
0
1
2
3
4
5
6
7
8
9
Gaussian integers
• Indian mathematician Brahmagupta treated the Pell’s equation X2 − 61 Y 2 = 1, corresponding to the ring is Z[√61]. Some results were presented to European community by Pierre Fermat in 1657.
182.5.3
Principal rings of quadratic integers
Unique factorization property is not always verified for rings of quadratic integers, as seen above for the case of Z[√−5]. However, as for every Dedekind domain, a ring of quadratic integers is a unique factorization domain if and only if it is a principal ideal domain. This occurs if and only if the class number of the corresponding quadratic field is one. The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are OQ(√D) for D = −1, −2, −3, −7, −11, −19, −43, −67, −163. This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner’s proof was not believed until Harold Stark gave a later proof in 1967. (See Stark–Heegner theorem.) This is a special case of the famous class number problem.
554
CHAPTER 182. QUADRATIC INTEGER
Eisenstein primes
There are many known positive integers D > 0, for which the ring of quadratic integers is a principal ideal ring. However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.
182.5.4
Euclidean rings of quadratic integers
See also: Euclidean domain § Norm-Euclidean fields When a ring of quadratic integers is a principal ideal domain, it is interesting to know if it is a Euclidean domain. This problem has been completely solved as follows. √ Equipped with the norm N (a + b D) = a2 − Db2 , as an Euclidean function, OQ(√D) is an Euclidean domain for negative D when D = −1, −2, −3, −7, −11, [5] and, for positive D, when D = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 (sequence A048981 in OEIS). There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function.[6] For negative D, a ring of quadratic integers is Euclidean if and only if the norm is a Euclidean function for it. It follows that, for
182.6. NOTES
555
Powers of the golden ratio
D = −19, −43, −67, −163, the four corresponding rings of quadratic integers are among the rare known examples of principal ideal domains that are not Euclidean domains. On the other hand, the generalized Riemann hypothesis implies that a ring of real quadratic integers that is a principal ideal domain is also a Euclidean domain for some Euclidean function.
182.6 Notes [1] Dedekind 1871, Supplement X, p. 447 [2] Bourbaki 1994, p. 99 [3] Dummit, pg. 229 [4] de Bruijn, N. G. (1981), “Algebraic theory of Penrose’s non-periodic tilings of the plane, I, II” (PDF), Indagationes mathematicae 43 (1): 39–66 [5] Dummit, pg. 272 [6] LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. pp. II:57,81. ISBN 978-0-486-42539-9. Zbl 1009.11001.
182.7 References • Bourbaki, Nicolas (1994), Elements of the history of mathematics, Berlin: Springer-Verlag, ISBN 978-3-54064767-6, MR 1290116. Translated from the original French by John Meldrum • Dedekind, Richard (1871), Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet (2 ed.), Vieweg. Retrieved 5. August 2009 • Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. • J.S. Milne. Algebraic Number Theory, Version 3.01, September 28, 2008. online lecture note
Chapter 183
Quadratic reciprocity “Law of reciprocity” redirects here. For the philosophical concept known as the “ethic of reciprocity”, see Golden Rule. In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem. One version of the law states that ( )( ) p−1 q−1 p q = (−1) 2 2 q p for p and q odd prime numbers, and
( ) p q
denoting the Legendre symbol.
This law, combined with the properties of the Legendre symbol, means that any Legendre symbol (a/p) can be calculated. This makes it possible to determine, for any quadratic equation, x2 ≡ a (mod p) , where p is an odd prime, if it has a solution. However, it does not provide any help at all for actually finding the solution. The solution can be found using quadratic residues. The theorem was conjectured by Euler and Legendre and first proved by Gauss.[1] He refers to it as the “fundamental theorem” in the Disquisitiones Arithmeticae and his papers, writing The fundamental theorem must certainly be regarded as one of the most elegant of its type. (Art. 151) Privately he referred to it as the “golden theorem.”[2] He published six proofs, and two more were found in his posthumous papers. There are now over 200 published proofs.[3] The first section of this article gives a special case of quadratic reciprocity that is representative of the general case. The second section gives the formulations of quadratic reciprocity found by Legendre and Gauss.
183.1 Motivating example Consider the polynomial f(n) = n2 − 5 and its values for n = 1, 2, 3, 4, ... The prime factorizations of these values are given as follows: A striking feature of the data is that with the exceptions of 2 and 5, the prime numbers that appear as factors are precisely those with final digit 1 or 9. Another way of phrasing this is that the primes p for which there exists an n such that n2 ≡ 5 (mod p) are precisely 2, 5, and those primes p that are ≡ 1 or 4 (mod 5). The law of quadratic reciprocity gives a similar characterization of prime divisors of f(n) = n2 − c for any integer c. 556
183.2. TERMINOLOGY, DATA, AND TWO STATEMENTS OF THE THEOREM
557
183.2 Terminology, data, and two statements of the theorem A quadratic residue (mod n) is any number congruent to a square (mod n). A quadratic nonresidue (mod n) is any number that is not congruent to a square (mod n). The adjective “quadratic” can be dropped if the context makes it clear that it is implied. When working modulo primes (as in this article), it is usual to treat zero as a special case. By doing so, the following statements become true: • Modulo a prime, there are an equal number of quadratic residues and nonresidues. • Modulo a prime, the product of two quadratic residues is a residue, the product of a residue and a nonresidue is a nonresidue, and the product of two nonresidues is a residue.
183.2.1
Table of quadratic residues
This table is complete for odd primes less than 50. To check whether a number m is a quadratic residue mod one of these primes p, find a ≡ m (mod p) and 0 ≤ a < p. If a is in row p, then m is a residue (mod p); if a is not in row p of the table, then m is a nonresidue (mod p). The quadratic reciprocity law is the statement that certain patterns found in the table are true in general. In this article, p and q always refer to distinct positive odd prime numbers.
183.2.2
−1 and the first supplement
First of all, for which prime numbers is −1 a quadratic residue? Examining the table, we find −1 in rows 5, 13, 17, 29, 37, and 41 but not in rows 3, 7, 11, 19, 23, 31, 43 or 47. (−1 ≡ 2 (mod 3), −1 ≡ 4 (mod 5), −1 ≡ 10 (mod 11), etc.) The former primes are all ≡ 1 (mod 4), and the latter are all ≡ 3 (mod 4). This leads to The first supplement to quadratic reciprocity:
congruence Thex2 ≡ −1
183.2.3
(mod p) if only and if solvable is p ≡ 1 (mod 4).
±2 and the second supplement
For which prime numbers is 2 a quadratic residue? Examining the table, we find 2 in rows 7, 17, 23, 31, 41, and 47, but not in rows 3, 5, 11, 13, 19, 29, 37, or 43. The former primes are all ≡ ±1 (mod 8), and the latter are all ≡ ±3 (mod 8). This leads to The second supplement to quadratic reciprocity:
congruence Thex2 ≡ 2 (mod p) if only and if solvable is p ≡ ±1 (mod 8). −2 is in rows 3, 11, 17, 19, 41, 43, but not in rows 5, 7, 13, 23, 29, 31, 37, or 47. The former are ≡ 1 or ≡ 3 (mod 8), and the latter are ≡ 5 or ≡ 7 (mod 8).
183.2.4
±3
3 is in rows 11, 13, 23, 37, and 47, but not in rows 5, 7, 17, 19, 29, 31, 41, or 43. The former are ≡ ±1 (mod 12) and the latter are all ≡ ±5 (mod 12). −3 is in rows 7, 13, 19, 31, 37, and 43 but not in rows 5, 11, 17, 23, 29, 41, or 47. The former are ≡ 1 (mod 3) and the latter ≡ 2 (mod 3).
558
CHAPTER 183. QUADRATIC RECIPROCITY
Since the only residue (mod 3) is 1, we see that −3 is a quadratic residue modulo every prime which is a residue (mod 3).
183.2.5
±5
5 is in rows 11, 19, 29, 31, and 41 but not in rows 3, 7, 13, 17, 23, 37, 43, or 47. The former are ≡ ±1 (mod 5) and the latter are ≡ ±2 (mod 5). Since the only residues (mod 5) are ±1, we see that 5 is a quadratic residue modulo every prime which is a residue (mod 5). −5 is in rows 3, 7, 23, 29, 41, 43, and 47 but not in rows 11, 13, 17, 19, 31, or 37. The former are ≡ 1, 3, 7, 9 (mod 20) and the latter are ≡ 11, 13, 17, 19 (mod 20).
183.2.6
Gauss’s version
The observations about −3 and +5 continue to hold: −7 is a residue (mod p) if and only if p is a residue (mod 7), −11 is a residue (mod p) if and only if p is a residue (mod 11), +13 is a residue (mod p) if and only if p is a residue (mod 13), ... The more complicated-looking rules for the quadratic characters of +3 and −5, which depend upon congruences (mod 12) and (mod 20) respectively, are simply the ones for −3 and +5 working with the first supplement. For example, for −5 to be a residue (mod p), either both 5 and −1 have to be residues (mod p) or they both have to be nonresidues: i.e., p has to be ≡ ±1 (mod 5) and ≡ 1 (mod 4), which is the same thing as p ≡ 1 or 9 (mod 20), or p has to be ≡ ±2 mod 5 and ≡ 3 mod 4, which is the same as p ≡ 3 or 7 (mod 20). See Chinese remainder theorem. The generalization of the rules for −3 and +5 is Gauss’s statement of quadratic reciprocity:
Ifq ≡ 1
(mod 4)then
congruence thex2 ≡ p Ifq ≡ 3
(mod q) if only and if solvable is x2 ≡ q
(mod p)but is,
(mod 4)then
congruence thex2 ≡ p
(mod q) if only and if solvable is x2 ≡ −q
(mod p)is.
These statements may be combined: Let q* = (−1)(q−1)/2 q. Then the congruence x2 ≡ p (mod q) is solvable if and only if x2 ≡ q* (mod p) is.
183.2.7
Table of quadratic character of primes
183.2.8
Legendre’s version
Another way to organize the data is to see which primes are residues mod which other primes, as illustrated in the above table. The entry in row p column q is R if q is a quadratic residue (mod p); if it is a nonresidue the entry is N. If the row, or the column, or both, are ≡ 1 (mod 4) the entry is blue or green; if both row and column are ≡ 3 (mod 4), it is yellow or orange. The blue and green entries are symmetric around the diagonal: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is R (resp N). The yellow and orange ones, on the other hand, are antisymmetric: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is N (resp R). This observation is Legendre’s statement of quadratic reciprocity:
183.3. CONNECTION WITH CYCLOTOMY
559
Ifp ≡ 1 (mod 4) or q ≡ 1 (mod 4)then both), (or x2 ≡ q
(mod p) if only and if solvable is x2 ≡ p
(mod q)solvable. is
Ifp ≡ q ≡ 3 (mod 4), then x2 ≡ q
(mod p) if only and if solvable is x2 ≡ p
(mod q)solvable. not is
It is a simple exercise to prove that Legendre’s and Gauss’s statements are equivalent – it requires no more than the first supplement and the facts about multiplying residues and nonresidues.
183.3 Connection with cyclotomy The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields. His proof was cast in modern form by later algebraic number theorists. This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity Robert Langlands formulated the Langlands program, which gives a conjectural vast generalization of class field theory. He wrote:[4] I confess that, as a student unaware of the history of the subject and unaware of the connection with cyclotomy, I did not find the law or its so-called elementary proofs appealing. I suppose, although I would not have (and could not have) expressed myself in this way that I saw it as little more than a mathematical curiosity, fit more for amateurs than for the attention of the serious mathematician that I then hoped to become. It was only in Hermann Weyl’s book on the algebraic theory of numbers[5] that I appreciated it as anything more.
183.4 History and alternative statements There are a number of ways to state the theorem. Keep in mind that Euler and Legendre did not have Gauss’s congruence notation, nor did Gauss have the Legendre symbol. In this article p and q always refer to distinct positive odd primes.
183.4.1
Fermat
Fermat proved[6] (or claimed to have proved)[7] a number of theorems about expressing a prime by a quadratic form:
p = x2 + y 2 if only and if p = 2 or p ≡ 1 (mod 4), p = x2 + 2y 2 if only and if p = 2 or p ≡ 1, 3 p = x + 3y if only and if p = 3 or p ≡ 1 2
2
(mod 8), (mod 3).
He did not state the law of quadratic reciprocity, although the cases −1, ±2, and ±3 are easy deductions from these and other of his theorems. He also claimed to have a proof that if the prime number p ends with 7, (in base 10) and the prime number q ends in 3, and p ≡ q ≡ 3 (mod 4), then
pq = x2 + 5y 2 . Euler conjectured, and Lagrange proved, that[8]
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CHAPTER 183. QUADRATIC RECIPROCITY
If p ≡ 1, 9 (mod 20) then p = x2 + 5y 2 , ifp, q ≡ 3, 7
(mod 20) then pq = x2 + 5y 2 .
Proving these and other statements of Fermat was one of the things that led mathematicians to the reciprocity theorem.
183.4.2
Euler
Translated into modern notation, Euler stated:[9] 1. If q ≡ 1 (mod 4) then q is a quadratic residue (mod p) if and only if p ≡ r (mod q), where r is a quadratic residue of q. 1. If q ≡ 3 (mod 4) then q is a quadratic residue (mod p) if and only if p ≡ ±b2 (mod 4q), where b is odd and not divisible by q. This is equivalent to quadratic reciprocity. He could not prove it, but he did prove the second supplement.[10]
183.4.3
Legendre and his symbol
Fermat proved that if p is a prime number and a is an integer,
ap ≡ a
(mod p).
Thus, if p does not divide a,
a(p−1)/2 ≡ ±1
(mod p).
Legendre[11] lets a and A represent positive primes ≡ 1 (mod 4) and b and B positive primes ≡ 3 (mod 4), and sets out a table of eight theorems that together are equivalent to quadratic reciprocity: He says that since expressions of the form N (c−1)/2 (mod c) (where N and c are relatively prime) will come up so often he will abbreviate them as: (
N c
) ≡ N (c−1)/2
(mod c) = ±1.
This is now known as the Legendre symbol, and an equivalent[12][13] definition is used today: for all integers a and all odd primes p ( ) 0 if a ≡ 0 (mod p) a = +1 if a ̸≡ 0 (mod p) integer some for and x, a ≡ x2 p −1 such no is there if x.
(mod p)
183.4. HISTORY AND ALTERNATIVE STATEMENTS
561
Legendre’s version of quadratic reciprocity ( ) ( ) + pq if p ≡ 1 (mod 4) or q ≡ 1 (mod 4) p ( ) = − q if p ≡ q ≡ 3 (mod 4) q p He notes that these can be combined: ( )( ) p−1 q−1 p q = (−1) 2 2 . q p A number of proofs, especially those based on Gauss’s Lemma,[14] explicitly calculate this formula. The supplementary laws using Legendre symbols (
) { p−1 −1 +1 = (−1) 2 = −1 p ( ) { p2 −1 2 +1 = (−1) 8 = −1 p
if p ≡ 1 (mod 4) if p ≡ 3 (mod 4) if p ≡ 1 or 7 (mod 8) if p ≡ 3 or 5 (mod 8)
Legendre’s attempt to prove reciprocity is based on a theorem of his:
Leta, b, and csatisfy that integers be gcd(a, b) = gcd(b, c) = gcd(c, a) = 1. of one least Atab, bc, ca < 0. (i.e. they don't all have the same sign) u2 ≡ −bc (mod a), v 2 ≡ −ca (mod b), and w2 ≡ −ab
(mod c)solvable. are
equation the Thenax2 + by 2 + cz 2 = 0 integers. in solution nontrivial a has E.g., Theorem I is handled by letting a ≡ 1 and b ≡ 3 (mod 4) be primes and assuming that ( ab ) = 1 and, contrary the theorem, that ( ab ) = −1. Then x2 + ay 2 − bz 2 = 0 has a solution, and taking congruences (mod 4) leads to a contradiction. This technique doesn't work for Theorem VIII. Let b ≡ B ≡ 3 (mod 4), and assume ( Bb ) = ( Bb ) = −1. Then if there is another prime p ≡ 1 (mod 4) such that ( pb ) = ( Bp ) = −1, the solvability of Bx2 + by 2 − pz 2 = 0 leads to a contradiction (mod 4). But Legendre was unable to prove there has to be such a prime p; he was later able to show that all that is required is “Legendre’s lemma":
Ifa ≡ 1
( ) a (mod 4) prime a exists there prime is β that such = −1, β
but he couldn't prove that either. Hilbert symbol (below) discusses how techniques based on the existence of solutions to ax2 + by 2 + cz 2 = 0 can be made to work.
183.4.4
Gauss
Gauss first proves[15] the supplementary laws. He sets[16] the basis for induction by proving the theorem for ±3 and ±5. Noting[17] that it is easier to state for −3 and +5 than it is for +3 or −5, he states[18] the general theorem in the form: If p is a prime of the form 4n + 1 then p, but if p is of the form 4n+3 then −p, is a quadratic residue (resp. nonresidue) of every prime, which, with a positive sign, is a residue (resp. nonresidue) of p.
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CHAPTER 183. QUADRATIC RECIPROCITY
In the next sentence, he christens it the “fundamental theorem” (Gauss never used the word “reciprocity”). Introducing the notation a R b (resp. a N b) to mean a is a quadratic residue (resp. nonresidue) (mod b), and letting a, a′, etc. represent positive primes ≡ 1 (mod 4) and b, b′, etc. positive primes ≡ 3 (mod 4), he breaks it out into the same 8 cases as Legendre: In the next Article he generalizes this to what are basically the rules for the Jacobi symbol (below). Letting A, A′, etc. represent any (prime or composite) positive numbers ≡ 1 (mod 4) and B, B′, etc. positive numbers ≡ 3 (mod 4): All of these cases take the form “if a prime is a residue (mod a composite), then the composite is a residue or nonresidue (mod the prime), depending on the congruences (mod 4)". He proves that these follow from cases 1) - 8). Gauss needed, and was able to prove,[19] a lemma similar to the one Legendre needed:
Ifp ≡ 1
√ (mod 8) prime odd an exists there then prime, is q < 2 p + 1 that such
( ) p = −1. q
The proof[20] of quadratic reciprocity is by complete induction (i.e. assuming it is true for all numbers less than n allows the deduction it is true for n) for each of the cases 1) to 8). Gauss’s version in Legendre symbols ( ) ( ) q if q ≡ 1 (mod 4) p p = ( −q ) q if q ≡ 3 (mod 4) p
These can be combined: q−1
Letq ∗ = (−1) 2 q words other (in |q ∗ | = |q| and q ∗ ≡ 1 ( ) ( ∗) p q Then = . q p
(mod 4)).
A number of proofs of the theorem, especially those based on Gauss sums,[21] or the splitting of primes in algebraic number fields,[22] derive this formula.
183.4.5
Other statements
Note that the statements in this section are equivalent to quadratic reciprocity: if, for example, Euler’s version is assumed, the Legendre-Gauss version can be deduced from it, and vice versa. Euler This form of quadratic reciprocity is derived from Euler’s work:[23]
Ifp ≡ ±q
( ) ( ) a a (mod 4a) then = . p q
Euler’s statement can be proved by using Gauss’s lemma. Gauss Gauss’s fourth proof[24] consists of proving this theorem (by comparing two formulas for the value of Gauss sums) and then restricting it to two primes: Let a, b, c, ... be unequal positive odd primes, whose product is n, and let m be the number of them that are ≡ 3 (mod 4); check whether n/a is a residue of a, whether n/b is a residue of b, .... The number of nonresidues found will be even when m ≡ 0, 1 (mod 4), and it will be odd if m ≡ 2, 3 (mod 4).
183.4. HISTORY AND ALTERNATIVE STATEMENTS
563
He gives the example. Let a = 3, b = 5, c = 7, and d = 11. Three of these, 3, 7, and 11 ≡ 3 (mod 4), so m ≡ 3 (mod 4). 5×7×11 R 3; 3×7×11 R 5; 3×5×11 R 7; and 3×5×7 N 11, so there are an odd number of nonresidues. Eisenstein Eisenstein[25] formulates this:
′
′
′
Ifp ̸= q, p ̸= q , p ≡ p
(mod 4), and q ≡ q
′
( )( ) ( ′)( ′) p q q p (mod 4) then = . q p q′ p′
Mordell Mordell[26] proved the following to be equivalent to quadratic reciprocity:
Leta, b, and c prime every for Then integers. be p divides that abc, ifax2 + by 2 + cz 2 ≡ 0 (mod 4abc/p) solution nontrivial a has does soax2 + by 2 + cz 2 ≡ 0 (mod 4abc).
183.4.6
Jacobi symbol
The Jacobi symbol is a generalization of the Legendre symbol; the main difference is that the bottom number has to be positive and odd, but does not have to be prime. If it is prime, the two symbols agree. It obeys the same rules of manipulation as the Legendre symbol. In particular (
) { −1 1 = (−1)(n−1)/2 = −1 n ( ) { 2 2 1 = (−1)(n −1)/8 = −1 n
if n ≡ 1 (mod 4) if n ≡ 3 (mod 4) if n ≡ 1 or 7 if n ≡ 3 or 5
(mod 8) (mod 8)
and if both numbers are positive and odd (this is sometimes called “Jacobi’s reciprocity law”): (m) n
= (−1)(m−1)(n−1)/4
(n) m
.
However, if the Jacobi symbol is +1 and the bottom number is composite, it does not necessarily mean that the top number is a quadratic residue of the bottom one. Gauss’s cases 9) - 14) above can be expressed in terms of Jacobi symbols: (
M p
(
)
(p−1)(M −1)/4
= (−1)
p M
) ,
and since p is prime the left hand side is a Legendre symbol, and we know whether M is a residue (mod p) or not. The formulas listed in the preceding section are true for Jacobi symbols as long as the symbols are defined. Euler’s formula may be written (a) m
( =
a m ± 4an
) where n and integer an is m ± 4an > 0.
564
CHAPTER 183. QUADRATIC RECIPROCITY 2 2 2 For example, ( 27 ) = ( 15 ) = ( 23 ) = ( 31 ) · · · = 1, and 2 is a residue mod the primes 7, 23 and 31: 32 ≡ 2 (mod 7), 52 ≡ 2 (mod 23), and 82 ≡ 2 (mod 31), but 2 is not a quadratic residue (mod 5), so it can't be one (mod 15). This is related to the problem a Legendre had: if we know that ( m ) = −1 , we know that a is a nonresidue modulo every prime in the arithmetic series m + 4a, m + 8a, ..., if there are any primes in this series, but that wasn't proved until decades[27] after Legendre.
Eisenstein’s formula requires relative primality conditions (which are true if the numbers are prime)
Ifa, b, a′ and b′ and odd and positive are gcd(a, b) = gcd(a′ , b′ ) = 1, then ifa ≡ a′
(mod 4) and b ≡ b′
183.4.7
Hilbert symbol
(mod 4),
( )( ) ( ′)( ′ ) b a b a = . b a b′ a′
The quadratic reciprocity law can be formulated in terms of the Hilbert symbol (a, b)v where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the archimedean one and the p-adic absolute values for primes p). The Hilbert symbol (a, b)v is 1 or −1. It is defined to be 1 if and only if the equation ax2 + by 2 = z 2 has a solution in the completion of the rationals at v other than x = y = z = 0 . The Hilbert reciprocity law states that (a, b)v , for fixed a and b and varying v, is 1 for all but finitely many v and the product of (a, b)v over all v is 1. (This formally resembles the residue theorem from complex analysis.) The proof of Hilbert reciprocity reduces to checking a few special cases, and the non-trivial cases turn out to be equivalent to the main law and the two supplementary laws of quadratic reciprocity for the Legendre symbol. There is no kind of reciprocity in the Hilbert reciprocity law; its name simply indicates the historical source of the result in quadratic reciprocity. Unlike quadratic reciprocity, which requires sign conditions (namely positivity of the primes involved) and a special treatment of the prime 2, the Hilbert reciprocity law treats all absolute values of the rationals on an equal footing. Therefore it is a more natural way of expressing quadratic reciprocity with a view towards generalization: the Hilbert reciprocity law extends with very few changes to all global fields and this extension can rightly be considered a generalization of quadratic reciprocity to all global fields.
183.5 Other rings There are also quadratic reciprocity laws in rings other than the integers.
183.5.1
Gaussian integers
In his second monograph on quartic reciprocity[28] Gauss stated quadratic reciprocity for the ring Z[i] of Gaussian integers, saying that it is a corollary of the biquadratic law in Z[i], but did not provide a proof of either theorem. Peter Gustav Lejeune Dirichlet[29] showed that the law in Z[i] can be deduced from the law for Z without using biquadratic reciprocity. For an odd Gaussian prime π and a Gaussian integer α, gcd(α, π) = 1, define the quadratic character for Z[i] by the formula [α] π
2
Nπ−1 2
≡α { =
(mod π)
+1 if gcd(α, π) = 1 integer Gaussian a is there and η that such α ≡ η 2 −1 if gcd(α, π) = 1 such no is there and η.
(mod π)
Let λ = a + b i and μ = c + d i be distinct Gaussian primes where a and c are odd and b and d are even. Then[30]
183.5. OTHER RINGS
[ ] [ ] λ µ = , µ λ 2
565
[ ] ( [ ] ) b i 1+i 2 2 = (−1) , and = , λ λ a+b
2
2
2
where ( ab ) is the Jacobi symbol for Z.
183.5.2
Eisenstein integers √
The ring of Eisenstein integers is Z[ω], where ω = −1+2 −3 = e Eisenstein integer and cubic reciprocity for definitions and notations).
2πi 3
is a cube root of 1. (See the articles on
For an Eisenstein prime π, Nπ ≠ 3 and an Eisenstein integer α, gcd(α, π) = 1, define the quadratic character for Z[ω] by the formula [α] π
2
Nπ−1
= ±1 ≡ α 2 (mod π) { +1 if gcd(α, π) = 1 integer Eisenstein an is there and η that such α ≡ η 2 = −1 if gcd(α, π) = 1 such no is there and η.
(mod π)
Let λ = a + b ω and μ = c + d ω be distinct Eisenstein primes where a and c are not divisible by 3 and b and d are divisible by 3. Eisenstein proved[31] [ ] [ ] Nλ−1 Nµ−1 µ λ 2 = (−1) 2 , µ 2 λ 2 where
( ab )
183.5.3
[
1−ω λ
] = 2
( ) ( [ ] ) a 2 2 = , and , 3 λ 2 Nλ
is the Jacobi symbol for Z.
Imaginary quadratic fields
The laws in Z[i] and Z[ω] are special cases of more general laws that hold for the ring of integers in any imaginary quadratic number field. Let k be an imaginary quadratic number field with ring of integers Ok . For a prime ideal p ⊂ Ok with odd norm Np and α ∈ Ok , define the quadratic character for Ok by the formula [ ] Np−1 α ≡α 2 (mod p) p 2 2 +1 if α ̸∈ p an is there and η ∈ Ok that such α − η ∈ p = −1 if α ̸∈ p such no is there and η 0 if α ∈ p, for an arbitrary ideal a ⊂ Ok factored into prime ideals a = p1 p2 . . . pn define [ ] [ ] [ ] [ ] α α α α = ... , a 2 p1 2 p2 2 pn 2 and for β ∈ Ok define [ ] [ ] α α = . β 2 βOk 2 Let {ω1 , ω2 } be an integral basis of Ok = Zω1 ⊕ Zω2 .
566
CHAPTER 183. QUADRATIC RECIPROCITY
For ν ∈ Ok with odd norm Nν, define (ordinary) integers a, b, c, d by the equations, νω1 = aω1 + bω2 νω2 = cω1 + dω2 and define a function χ(ν) where ν has odd norm by 2
χ(ν) = i(b
−a+2)c+(a2 −b+2)d+ad
.
If m = Nμ and n = Nν are both odd, Herglotz proved[32] [ ] [ ] n−1 m−1 m−1 n−1 µ ν = (−1) 2 2 χ(µ)m 2 χ(ν)−n 2 . ν µ 2 2
Also, if µ ≡ µ′ (mod 4) and ν ≡ ν ′ (mod 4) [33] [ ] [ ] [ ] [ ] µ ν µ′ ν′ = . ν µ 2 ν′ µ′ 2 2
183.5.4
2
Polynomials over a finite field
Let F be a finite field with q = pn elements, where p is an odd prime number and n is positive, and let F[x] be the ring of polynomials in one variable with coefficients in F. If f, g ∈ F[x] and f is irreducible, monic, and has positive degree, define the quadratic character ( fg ) for F[x] in the usual manner: 2 ( ) +1 if gcd(f, g) = 1 are there and h, k ∈ F[x] that such g − h = kf g = −1 if gcd(f, g) = 1 and g square a not is (mod f ) f 0 if gcd(f, g) = ̸ 1. If f = f1 f2 . . . fn is a product of monic irreducibles let ( ) ( ) ( )( ) g g g g = ... . f f1 f2 fn Dedekind[34] proved that if f, g ∈ F[x] are monic and have positive degrees, ( )( ) q−1 g f = (−1) 2 (deg f )(deg g) . f g
183.6 Higher powers The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led 19th century mathematicians, including Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet, Carl Gustav Jakob Jacobi, Gotthold Eisenstein, Richard Dedekind, Ernst Kummer, and David Hilbert to the study of general algebraic number fields and their rings of integers;[35] specifically Kummer invented ideals in order to state and prove higher reciprocity laws. The ninth in the list of 23 unsolved problems which David Hilbert proposed to the Congress of Mathematicians in 1900 asked for the “Proof of the most general reciprocity law [f]or an arbitrary number field”.[36] In 1923 Emil Artin, building upon work by Philipp Furtwängler, Teiji Takagi, Helmut Hasse and others, discovered a general theorem for which all known reciprocity laws are special cases; he proved it in 1927.[37] The links below provide more detailed discussions of these theorems.
183.7. SEE ALSO
567
183.7 See also • Euler’s criterion • Zolotarev’s lemma • Proofs of quadratic reciprocity • Cubic reciprocity • Quartic reciprocity • Eisenstein reciprocity • Artin reciprocity
183.8 Notes [1] Gauss, DA § 4, arts 107–150 [2] E.g. in his mathematical diary entry for April 8, 1796 (the date he first proved quadratic reciprocity). See facsimile page from Felix Klein’s Development of Mathematics in the 19th century [3] See F. Lemmermeyer’s chronology and bibliography of proofs in the external references [4] http://www.math.duke.edu/langlands/Three.pdf [5] http://www.amazon.com/Algebraic-Theory-Numbers-Hermann-Weyl/dp/0691059179 [6] Lemmermeyer, pp. 2–3 [7] Gauss, DA, art. 182 [8] Lemmermeyer, p. 3 [9] Lemmermeyer, p. 5, Ireland & Rosen, pp. 54, 61 [10] Ireland & Rosen, pp. 69–70. His proof is based on what are now called Gauss sums. [11] This section is based on Lemmermeyer, pp. 6–8 [12] The equivalence is Euler’s criterion [13] The analogue of Legendre’s original definition is used for higher-power residue symbols [14] E.g. Kronecker’s proof (Lemmermeyer, ex. p. 31, 1.34) is to use Gauss’s lemma to establish that q−1 p−1 ( ) ) 2 2 ( ∏ ∏ p k i = sgn − q p q i=1
k=1
and then switch p and q. [15] Gauss, DA, arts 108–116 [16] Gauss, DA, arts 117–123 [17] Gauss, DA, arts 130 [18] Gauss, DA, Art 131 [19] Gauss, DA, arts. 125–129 [20] Gauss, DA, arts 135–144 √ [21] Because the basic Gauss sum equals q ∗ . √ [22] Because the quadratic field Q( q ∗ ) is a subfield of the cyclotomic field Q(e2πi/q ) [23] Ireland & Rosen, pp 60–61.
568
CHAPTER 183. QUADRATIC RECIPROCITY
[24] Gauss, “Summierung gewisser Reihen von besonderer Art”, reprinted in Untersuchumgen uber hohere Arithmetik, pp.463– 495 [25] Lemmermeyer, Th. 2.28, pp 63–65 [26] Lemmermeyer, ex. 1.9, p. 28 [27] By Peter Gustav Lejeune Dirichlet in 1837 [28] Gauss, BQ § 60 [29] Dirichlet’s proof is in Lemmermeyer, Prop. 5.1 p.154, and Ireland & Rosen, ex. 26 p. 64 [30] Lemmermeyer, Prop. 5.1, p. 154 [31] Lemmermeyer, Thm. 7.10, p. 217 [32] Lemmermeyer, Thm 8.15, p.256 ff [33] Lemmermeyer Thm. 8.18, p. 260 [34] Bach & Shallit, Thm. 6.7.1 [35] Lemmermeyer, p. 15, and Edwards, pp.79–80 both make strong cases that the study of higher reciprocity was much more important as a motivation than Fermat’s Last Theorem was [36] Lemmermeyer, p. viii [37] Lemmermeyer, p. ix ff
183.9 References The Disquisitiones Arithmeticae has been translated (from Latin) into English and German. The German edition includes all of Gauss’s papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Footnotes referencing the Disquisitiones Arithmeticae are of the form “Gauss, DA, Art. n". • Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9 • Gauss, Carl Friedrich; Maser, Hermann (translator into German) (1965), Untersuchungen über höhere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8 The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form “Gauss, BQ, § n". • Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6 • Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7 These are in Gauss’s Werke, Vol II, pp. 65–92 and 93–148. German translations are in pp. 511–533 and 534–586 of Untersuchungen über höhere Arithmetik. Every textbook on elementary number theory (and quite a few on algebraic number theory) has a proof of quadratic reciprocity. Two are especially noteworthy: Franz Lemmermeyer’s Reciprocity Laws: From Euler to Eisenstein has many proofs (some in exercises) of both quadratic and higher-power reciprocity laws and a discussion of their history. Its immense bibliography includes literature citations for 196 different published proofs for the quadratic reciprocity law. Kenneth Ireland and Michael Rosen's A Classical Introduction to Modern Number Theory also has many proofs of quadratic reciprocity (and many exercises), and covers the cubic and biquadratic cases as well. Exercise 13.26 (p.202) says it all
183.10. EXTERNAL LINKS
569
Count the number of proofs to the law of quadratic reciprocity given thus far in this book and devise another one. • Bach, Eric; Shallit, Jeffrey (1966), Algorithmic Number Theory (Vol I: Efficient Algorithms), Cambridge: The MIT Press, ISBN 0-262-02405-5 • Edwards, Harold (1977), Fermat’s Last Theorem, New York: Springer, ISBN 0-387-90230-9 • Lemmermeyer, Franz (2000), Reciprocity Laws, Springer Monographs in Mathematics, Berlin: SpringerVerlag, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696 • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (second edition), New York: Springer, ISBN 0-387-97329-X
183.10 External links • Hazewinkel, Michiel, ed. (2001), “Quadratic reciprocity law”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Quadratic Reciprocity Theorem from MathWorld • A play comparing two proofs of the quadratic reciprocity law • A proof of this theorem at PlanetMath • A different proof at MathPages • F. Lemmermeyer’s chronology and bibliography of proofs of the Quadratic Reciprocity Law (233 proofs)
570
CHAPTER 183. QUADRATIC RECIPROCITY
Part of Article 131 in the first edition (1801) of the Disquisitiones, listing the 8 cases of quadratic reciprocity
Chapter 184
Quadratically closed field In mathematics, a quadratically closed field is a field in which every element of the field has a square root in the field.[1][2]
184.1 Examples • The field of complex numbers is quadratically closed; more generally, any algebraically closed field is quadratically closed. • The field of real numbers is not quadratically closed as it does not contain a square root of −1. • The union of the finite fields F52n for n ≥ 0 is quadratically closed but not algebraically closed.[3] • The field of constructible numbers is quadratically closed but not algebraically closed.[4]
184.2 Properties • A field is quadratically closed if and only if it has universal invariant equal to 1. • Every quadratically closed field is a Pythagorean field but not conversely (for example, R is Pythagorean); however, every non-formally real Pythagorean field is quadratically closed.[2] • A field is quadratically closed if and only if its Witt–Grothendieck ring is isomorphic to Z under the dimension mapping.[3] • A formally real Euclidean field E is not quadratically closed (as −1 is not a square in E) but the quadratic extension E(√−1) is quadratically closed.[4] • Let E/F be a finite extension where E is quadratically closed. Either −1 is a square in F and F is quadratically closed, or −1 is not a square in F and F is Euclidean. This “going-down theorem” may be deduced from the Diller–Dress theorem.[5]
184.3 Quadratic closure A quadratic closure of a field F is a quadratically closed field which embeds in any other quadratically closed field containing F. A quadratic closure for any given F may be constructed as a subfield of the algebraic closure F alg of F, as the union of all quadratic extensions of F in F alg .[4] 571
572
184.3.1
CHAPTER 184. QUADRATICALLY CLOSED FIELD
Examples
• The quadratic closure of R is C.[4] • The quadratic closure of F5 is the union of the F52n .[4] • The quadratic closure of Q is the field of constructible numbers.
184.4 References [1] Lam (2005) p. 33 [2] Rajwade (1993) p. 230 [3] Lam (2005) p. 34 [4] Lam (2005) p. 220 [5] Lam (2005) p.270
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
Chapter 185
Quartic reciprocity Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word “reciprocity” comes from the form of some of these theorems, in that they relate the solvability of the congruence x4 ≡ p (mod q) to that of x4 ≡ q (mod p).
185.1 History Euler made the first conjectures about biquadratic reciprocity.[1] Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler’s conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He said[2] that a third monograph would be forthcoming with the proof of the general theorem, but it never appeared. Jacobi presented proofs in his Königsberg lectures of 1836–37.[3] The first published proofs were by Eisenstein.[4][5][6][7] Since then a number of other proofs of the classical (Gaussian) version have been found,[8] as well as alternate statements. Lemmermeyer states that there has been an explosion of interest in the rational reciprocity laws since the 1970s.[A][9]
185.2 Integers A quartic or biquadratic residue (mod p) is any number congruent to the fourth power of an integer (mod p). If x4 ≡ a (mod p) does not have an integer solution, a is a quartic or biquadratic nonresidue (mod p).[10] As is often the case in number theory, it is easiest to work modulo prime numbers, so in this section all moduli p, q, etc., are assumed to positive, odd primes.[10]
185.2.1
Gauss
The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 3 (mod 4) then every quadratic residue (mod q) is also a biquadratic residue (mod q). The first supplement of quadratic reciprocity states that −1 is a quadratic nonresidue (mod q), so that for any integer x, one of x and −x is a quadratic residue and the other one is a nonresidue. Thus, if r ≡ a2 (mod q) is a quadratic residue, then if a ≡ b2 is a residue, r ≡ a2 ≡ b4 (mod q) is a biquadratic residue, and if a is a nonresidue, −a is a residue, −a ≡ b2 , and again, r ≡ (−a)2 ≡ b4 (mod q) is a biquadratic residue.[11] Therefore, the only interesting case is when the modulus p ≡ 1 (mod 4). Gauss proved[12] that if p ≡ 1 (mod 4) then the nonzero residue classes (mod p) can be divided into four sets, each containing (p−1)/4 numbers. Let e be a quadratic nonresidue. The first set is the quartic residues; the second one is e times the numbers in the first set, the third is e2 times the numbers in the first set, and the fourth one is e3 times the numbers in the first set. Another way to describe this division is to let g be a primitive root (mod p); then the first set 573
574
CHAPTER 185. QUARTIC RECIPROCITY
is all the numbers whose indices with respect to this root are ≡ 0 (mod 4), the second set is all those whose indices are ≡ 1 (mod 4), etc.[13] In the vocabulary of group theory, the first set is a subgroup of index 4 (of the multiplicative group Z/pZ× ), and the other three are its cosets. The first set is the biquadratic residues, the third set is the quadratic residues that are not quartic residues, and the second and fourth sets are the quadratic nonresidues. Gauss proved that −1 is a biquadratic residue if p ≡ 1 (mod 8) and a quadratic, but not biquadratic, residue, when p ≡ 5 (mod 8).[14] 2 is a quadratic residue mod p if and only if p ≡ ±1 (mod 8). Since p is also ≡ 1 (mod 4), this means p ≡ 1 (mod 8). Every such prime is the sum of a square and twice a square.[15] Gauss proved[14] Let q = a2 + 2b2 ≡ 1 (mod 8) be a prime number. Then
2 is a biquadratic residue (mod q) if and only if a ≡ ±1 (mod 8), and 2 is a quadratic, but not a biquadratic, residue (mod q) if and only if a ≡ ±3 (mod 8). Every prime p ≡ 1 (mod 4) is the sum of two squares.[16] If p = a2 + b2 where a is odd and b is even, Gauss proved[17] that 2 belongs to the first (respectively second, third, or fourth) class defined above if and only if b ≡ 0 (resp. 2, 4, or 6) (mod 8). The first case of this is one of Euler’s conjectures: 2 is a biquadratic residue of a prime p ≡ 1 (mod 4) if and only if p = a2 + 64b2 .
185.2.2
Dirichlet
For an odd prime number p and a quadratic residue a (mod p), Euler’s criterion states that a p−1 p ≡ 1 (mod 4), a 4 ≡ ±1 (mod p).
p−1 2
≡ 1 (mod p), so if ( )
Define the rational quartic residue symbol for prime p ≡ 1 (mod 4) and quadratic residue a (mod p) as ( ) ±1 ≡ a
p−1 4
(mod p). It is easy to prove that a is a biquadratic residue (mod p) if and only if
a p
a p
= 4
= 1. 4
Dirichlet[18] simplified Gauss’s proof of the biquadratic character of 2 (his proof only requires quadratic reciprocity for the integers) and put the result in the following form: Let p = a2 + b2 ≡ 1 (mod 4) be prime, and let i ≡ b/a (mod p). Then ( ) ≡i
2 p
ab 2
(mod p). (Note that i2 ≡ −1 (mod p).)
4
In fact,[19] let p = a2 + b2 = c2 + 2d2 = e2 − 2f 2 ≡ 1 (mod 8) be prime, and assume a is odd. Then ( )
( ) b 4
2 p
2 c
= (−1) =
)
( n+ d 2
= (−1)
=
−2 e
, where ( xq ) is the ordinary Legendre symbol.
4
Going beyond the character of 2, let the prime p = a2 + b2 where b is even, and let q be a prime such that ( pq ) = 1. ∗
Quadratic reciprocity says that ( qp ) = 1, where q ∗ = (−1) )
(
(
q∗ p
= 4
) σ(b+σ) q
. This implies[21] that
q−1 2
q. Let σ2 ≡ p (mod q). Then[20]
185.2. INTEGERS (
q∗ p
) 4
575
b ≡ 0 (mod q); ( ) 2 = 1 if and only if a ≡ 0 (mod q) and q = 1; ( ) a ≡ µb, µ2 + 1 ≡ λ2 (mod q), and λ(λ+1) = 1.
or or
q
[22]
The first few examples are: (
) −3 p ( )4 5 p 4 ( ) −7 p 4 ( ) −11 p ( )4 13 p ( )4 17 p 4
= 1 if and only if
b ≡ 0 (mod 3)
= 1 if and only if
b ≡ 0 (mod 5)
= 1 if and only if
ab ≡ 0 (mod 7)
= 1 if and only if
b(b2 − 3a2 ) ≡ 0 (mod 11)
= 1 if and only if
b(b2 − 3a2 ) ≡ 0 (mod 13)
= 1 if and only if
ab(b2 − a2 ) ≡ 0 (mod 17).
Euler had conjectured the rules for 2, −3 and 5, but did not prove any of them. Dirichlet[23] also proved that if p ≡ 1 (mod 4) is prime and ( 17 p ) = 1 then (
17 p
) ( 4
p 17
) 4
{ +1 if and only if p = x2 + 17y 2 = −1 if and only if 2p = x2 + 17y 2
This has been extended from 17 to 17, 73, 97, and 193 by Brown and Lehmer.[24]
185.2.3
Burde
There are a number of equivalent ways of stating Burde’s rational biquadratic reciprocity law. They all assume that p = a2 + b2 and q = c2 + d2 are primes where b and d are even, and that ( pq ) = 1. Gosset’s version is[9] ( ) q−1 ( ) 4 q a/b − c/d ≡ p a/b + c/d
(mod q).
4
Letting i2 ≡ −1 (mod p) and j 2 ≡ −1 (mod q), Frölich’s law is[25] ( ) ( ) ( ) ( ) p a + bj c + di q = = . p q q p 4
4
Burde stated his in the form:[26][27][28] ( ) ( ) ( ) q p ac − bd = . p q q 4
4
[29]
Note that (
ac + bd p
)
( )( ) p ac − bd = . q p
576
CHAPTER 185. QUARTIC RECIPROCITY
185.2.4
Miscellany
Let p ≡ q ≡ 1 (mod 4) be primes and assume ( pq ) = 1 . Then e2 = p f 2 + q g2 has non-trivial integer solutions, and[30] ( ) ( ) ( ) fg p q −1 2 = (−1) . q p e 4
4
Let p ≡ q ≡ 1 (mod 4) be primes and assume p = r2 + q s2 . Then[31] ( ) ( ) ( )s p q 2 . = q p q 4
4
Let p = 1 + 4x2 be prime, let a be any odd number that divides x, and let a∗ = (−1) residue (mod p).
a−1 2
a. Then[32] a* is a biquadratic
Let p = a2 + 4b2 = c2 + 2d2 ≡ 1 (mod 8) be prime. Then[33] all the divisors of c4 − p a2 are biquadratic residues (mod p). The same is true for all the divisors of d4 − p b2 .
185.3 Gaussian integers 185.3.1
Background
In his second monograph on biquadratic reciprocity Gauss displays some examples and makes conjectures that imply the theorems listed above for the biquadratic character of small primes. He makes some general remarks, and admits there is no obvious general rule at work. He goes on to say The theorems on biquadratic residues gleam with the greatest simplilcity and genuine beauty only when the field of arithmetic is extended to imaginary numbers, so that without restriction, the numbers of the form a + bi constitute the object of study ... we call such numbers integral complex numbers.[34] [bold in the original] These numbers are now called the ring of Gaussian integers, denoted by Z[i]. Note that i is a fourth root of 1. In a footnote he adds The theory of cubic residues must be based in a similar way on a consideration of numbers of the form a + bh where h is an imaginary root of the equation h3 = 1 ... and similarly the theory of residues of higher powers leads to the introduction of other imaginary quantities.[35] The numbers built up from a cube root of unity are now called the ring of Eisenstein integers. The “other imaginary quantities” needed for the “theory of residues of higher powers” are the rings of integers of the cyclotomic number fields; the Gaussian and Eisenstein integers are the simplest examples of these.
185.3.2
Facts and terminology
Gauss develops the arithmetic theory of the “integral complex numbers” and shows that it is quite similar to the arithmetic of ordinary integers.[36] This is where the terms unit, associate, norm, and primary were introduced into mathematics. The units are the numbers that divide 1.[37] They are 1, i, −1, and −i. They are similar to 1 and −1 in the ordinary integers, in that they divide every number. The units are the powers of i. Given a number λ = a + bi, its conjugate is a − bi and its associates are the four numbers[37]
185.3. GAUSSIAN INTEGERS
577
λ = +a + bi iλ = −b + ai −λ = −a − bi −iλ = +b − ai The norm of λ = a + bi is the number Nλ = a2 + b2 . If λ and μ are two Gaussian integers, Nλμ = Nλ Nμ; in other words, the norm is multiplicative.[37] The norm of zero is zero, the norm of any other number is a positive integer. ε is a unit if and only if Nε = 1. Gauss proves that Z[i] is a unique factorization domain and shows that the primes fall into three classes:[38] • 2 is a special case: 2 = i3 (1 + i)2 . It is the only prime in Z divisible by the square of a prime in Z[i]. In algebraic number theory, 2 is said to ramify in Z[i]. • Positive primes in Z ≡ 3 (mod 4) are also primes in Z[i]. In algebraic number theory, these primes are said to remain inert in Z[i]. • Positive primes in Z ≡ 1 (mod 4) are the product of two conjugate primes in Z[i]. In algebraic number theory, these primes are said to split in Z[i]. Thus, inert primes are 3, 7, 11, 19, ... and a factorization of the split primes is 5 = (2 + i) × (2 − i), 13 = (2 + 3i) × (2 − 3i), 17 = (4 + i) × (4 − i), 29 = (2 + 5i) × (2 − 5i), ... The associates and conjugate of a prime are also primes. Note that the norm of an inert prime q is Nq = q2 ≡ 1 (mod 4); thus the norm of all primes other than 1 + i and its associates is ≡ 1 (mod 4). Gauss calls a number in Z[i] odd if its norm is an odd integer.[39] Thus all primes except 1 + i and its associates are odd. The product of two odd numbers is odd and the conjugate and associates of an odd number are odd. In order to state the unique factorization theorem, it is necessary to have a way of distinguishing one of the associates of a number. Gauss defines[40] an odd number to be primary if it is ≡ 1 (mod (1 + i)3 ). It is straightforward to show that every odd number has exactly one primary associate. An odd number λ = a + bi is primary if a + b ≡ a − b ≡ 1 (mod 4); i.e., a ≡ 1 and b ≡ 0, or a ≡ 3 and b ≡ 2 (mod 4).[41] The product of two primary numbers is primary and the conjugate of a primary number is also primary. The unique factorization theorem[42] for Z[i] is: if λ ≠ 0, then
λ = iµ (1 + i)ν π1α1 π2α2 π3α3 . . . where 0 ≤ μ ≤ 3, ν ≥ 0, the πis are primary primes and the αis ≥ 1, and this representation is unique, up to the order of the factors. The notions of congruence[43] and greatest common divisor[44] are defined the same way in Z[i] as they are for the ordinary integers Z. Because the units divide all numbers, a congruence (mod λ) is also true modulo any associate of λ, and any associate of a GCD is also a GCD.
185.3.3
Quartic residue character
Gauss proves the analogue of Fermat’s theorem: if α is not divisible by an odd prime π, then[45]
αN π−1 ≡ 1 (mod π)
578
CHAPTER 185. QUARTIC RECIPROCITY
Since Nπ ≡ 1 (mod 4), α
N π−1 4
makes sense, and α
N π−1 4
≡ ik (mod π) for a unique unit ik .
This unit is called the quartic or biquadratic residue character of α (mod π) and is denoted by[46][47] [α] π
= ik ≡ α
N π−1 4
(mod π).
It has formal properties similar to those of the Legendre symbol.[48] The congruence x4 ≡ α (mod π) is solvable in Z[i] if and only if [
αβ π
]
[α] π
= 1. [49]
[ ][ ] α β = π π
[ ] α π
[ ] =
α π
where the bar denotes complex conjugation. [ ] α π
if π and θ are associates, [ ] if α ≡ β (mod π),
α π
[ ] =
α θ
[ ] =
β π
The biquadratic character can be extended to odd composite numbers in the “denominator” in the same way the Legendre symbol is generalized into the Jacobi symbol. As in that case, if the “denominator” is composite, the symbol can equal one without the conguence being solvable: [α] λ
[ =
α π1
]α1 [
α π2
]α2
. . . where λ = π1α1 π2α2 π3α3 . . .
If a and b are ordinary integers, a ≠ 0, |b| > 1, gcd(a, b) = 1, then[50]
185.3.4
[a] b
= 1.
Statements of the theorem
Gauss stated the law of biquadratic reciprocity in this form:[2][51] Let π and θ be distinct primary primes of Z[i]. Then [ ] if either π or θ or both are ≡ 1 (mod 4), then
π θ
=
[ ] if both π and θ are ≡ 3 + 2i (mod 4), then
π θ
=−
[θ] π
[θ] π
, but
.
Just as the quadratic reciprocity law for the Legendre symbol is also true for the Jacobi symbol, the requirement that the numbers be prime is not needed; it suffices that they be odd relatively prime nonunits.[52] Probably the most well-known statement is: Let π and θ be primary relatively prime nonunits. Then[53] [ ][ ] −1 N π−1 N θ−1 π θ 4 = (−1) 4 . θ π
185.4. SEE ALSO
579
There are supplementary theorems[54][55] for the units and the half-even prime 1 + i. if π = a + bi is a primary prime, then [ ] [ ] a−b−1−b2 i 1+i − a−1 4 =i 2 , =i , π π and thus [
] [ ] a−1 b −1 2 2 = (−1) = i− 2 . , π π
Also, if π = a + bi is a primary prime, and b ≠ 0 then[56] [ ] π π
[ =
] −2 π
(−1)
a2 −1 8
(if b = 0 the symbol is 0).
Jacobi defined π = a + bi to be primary if a ≡ 1 (mod 4). With this normalization, the law takes the form[57] Let α = a + bi and β = c + di where a ≡ c ≡ 1 (mod 4) and b and d are even be relatively prime nonunits. Then [ ] [ ]−1 bd α β = (−1) 4 β α The following version was found in Gauss’s unpublished manuscripts.[58] Let α = a + 2bi and β = c + 2di where a and c are odd be relatively prime nonunits. Then [ ] [ ]−1 c−1 a−1 α β = (−1)bd+ 2 d+ 2 b , β α
[
] b(a−3b) a2 −1 1+i =i 2 − 8 α
The law can be stated without using the concept of primary: If λ is odd, let ε(λ) be the unique unit congruent to λ (mod (1 + i)3 ); i.e., ε(λ) = ik ≡ λ (mod 2 + 2i), where 0 ≤ k ≤ 3. Then[59] for odd and relatively prime α and β, neither one a unit, [ ] [ ]−1 N β−1 N α−1 N β−1 N α−1 α β 4 = (−1) 4 ϵ(α) 4 ϵ(β) 4 β α For odd λ, let λ∗ = (−1)
N λ−1 4
[ ] [ ∗] µ λ = . µ λ
185.4 See also • Quadratic reciprocity • Cubic reciprocity • Eisenstein reciprocity • Artin reciprocity
λ. Then if λ and μ are relatively prime nonunits, Eisenstein proved[60]
580
CHAPTER 185. QUARTIC RECIPROCITY
185.5 Notes • A.^ Here, “rational” means laws that are stated in terms of ordinary integers rather than in terms of the integers of some algebraic number field.
185.6 References [1] Euler, Tractatus, § 456 [2] Gauss, BQ, § 67 [3] Lemmermeyer, p. 200 [4] Eisenstein, Lois de reciprocite [5] Eisenstein, Einfacher Beweis ... [6] Eisenstein, Application de l'algebre ... [7] Eisenstein, Beitrage zur Theorie der elliptischen ... [8] Lemmermeyer, pp. 199–202 [9] Lemmermeyer, p. 172 [10] Gauss, BQ § 2 [11] Gauss, BQ § 3 [12] Gauss, BQ §§ 4–7 [13] Gauss, BQ § 8 [14] Gauss, BQ § 10 [15] Gauss, DA Art. 182 [16] Gauss, DA, Art. 182 [17] Gauss BQ §§ 14–21 [18] Dirichlet, Demonstration ... [19] Lemmermeyer, Prop. 5.4 [20] Lemmermeyer, Prop. 5.5 [21] Lemmermeyer, Ex. 5.6 [22] Lemmmermeyer, pp.159, 190 [23] Dirichlet, Untersuchungen ... [24] Lemmermeyer, Ex. 5.19 [25] Lemmermeyer, p. 173 [26] Lemmermeyer, p. 167 [27] Ireland & Rosen pp.128–130 [28] Burde, K. (1969). “Ein rationales biquadratisches Reziprozitätsgesetz”. J. Reine Angew. Math. (in German) 235: 175–184. Zbl 0169.36902. [29] Lemmermeyer, Ex. 5.13 [30] Lemmermeyer, Ex. 5.5 [31] Lemmermeyer, Ex. 5.6, credited to Brown
185.7. LITERATURE
581
[32] Lemmermeyer, Ex. 6.5, credited to Sharifi [33] Lemmermeyer, Ex. 6.11, credited to E. Lehmer [34] Gauss, BQ, § 30, translation in Cox, p. 83 [35] Gauss, BQ, § 30, translation in Cox, p. 84 [36] Gauss, BQ, §§ 30–55 [37] Gauss, BQ, § 31 [38] Gauss, BQ, §§ 33–34 [39] Gauss, BQ, § 35. He defines “halfeven” numbers as those divisible by 1 + i but not by 2, and “even” numbers as those divisible by 2. [40] Gauss, BQ, § 36 [41] Ireland & Rosen, Ch. 9.7 [42] Gauss, BQ, § 37 [43] Gauss, BQ, §§ 38–45 [44] Gauss, BQ, §§ 46–47 [45] Gauss, BQ, § 51 [46] Gauss defined the character as the exponent k rather than the unit ik ; also, he had no symbol for the character. [47] There is no standard notation for higher residue characters in different domains (see Lemmermeyer, p. xiv); this article follows Lemmermeyer, chs. 5–6 [48] Ireland & Rosen, Prop 9.8.3 [49] Gauss, BQ, § 61 [50] Ireland & Rosen, Prop. 9.8.3, Lemmermeyer, Prop 6.8 [51] proofs are in Lemmermeyer, chs. 6 and 8, Ireland & Rosen, ch. 9.7–9.10 [52] Lemmermeyer, Th. 69. [53] Lemmermeyer, ch. 6, Ireland & Rosen ch. 9.7–9.10 [54] Lemmermeyer, Th. 6.9; Ireland & Rosen, Ex. 9.32–9.37 [55] Gauss proves the law for 1 + i in BQ, §§ 68–76 [56] Ireland & Rosen, Ex. 9.30; Lemmermeyer, Ex. 6.6, where Jacobi is credited [57] Lemmermeyer, Th. 6.9 [58] Lemmermeyer, Ex. 6.17 [59] Lemmermeyer, Ex. 6.18 and p. 275 [60] Lemmermeyer, Ch. 8.4, Ex. 8.19
185.7 Literature The references to the original papers of Euler, Dirichlet, and Eisenstein were copied from the bibliographies in Lemmermeyer and Cox, and were not used in the preparation of this article.
582
185.7.1
CHAPTER 185. QUARTIC RECIPROCITY
Euler
• Euler, Leonhard (1849), Tractatus de numeroroum doctrina capita sedecim quae supersunt, Comment. Arithmet. 2 This was actually written 1748–1750, but was only published posthumously; It is in Vol V, pp. 182–283 of • Euler, Leonhard (1911–1944), Opera Omnia, Series prima, Vols I–V, Leipzig & Berlin: Teubner
185.7.2
Gauss
The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form “Gauss, BQ, § n". Footnotes referencing the Disquisitiones Arithmeticae are of the form “Gauss, DA, Art. n". • Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6 • Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7 These are in Gauss’s Werke, Vol II, pp. 65–92 and 93–148 German translations are in pp. 511–533 and 534–586 of the following, which also has the Disquisitiones Arithmeticae and Gauss’s other papers on number theory. • Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8
185.7.3
Eisenstein
• Eisenstein, Ferdinand Gotthold (1844), Lois de réciprocité, J. Reine Angew. Math. 28, pp. 53–67 (Crelle’s Journal) • Eisenstein, Ferdinand Gotthold (1844), Einfacher Beweis und Verallgemeinerung des Fundamentaltheorems für die biquadratischen Reste, J. Reine Angew. Math. 28 pp. 223–245 (Crelle’s Journal) • Eisenstein, Ferdinand Gotthold (1845), Application de l'algèbre à l'arithmétique transcendante, J. Reine Angew. Math. 29 pp. 177–184 (Crelle’s Journal) • Eisenstein, Ferdinand Gotthold (1846), Beiträge zur Theorie der elliptischen Funktionen I: Ableitung des biquadratischen Fundalmentaltheorems aus der Theorie der Lemniskatenfunctionen, nebst Bemerkungen zu den Multiplications- und Transformationsformeln, J. Reine Angew. Math. 30 pp. 185–210 (Crelle’s Journal) These papers are all in Vol I of his Werke.
185.7.4
Dirichlet
• Dirichlet, Pierre Gustave LeJeune (1832), Démonstration d'une propriété analogue à la loi de Réciprocité qui existe entre deux nombres premiers quelconques, J. Reine Angew. Math. 9 pp. 379–389 (Crelle’s Journal) • Dirichlet, Pierre Gustave LeJeune (1833), Untersuchungen über die Theorie der quadratischen Formen, Abh. Königl. Preuss. Akad. Wiss. pp. 101–121 both of these are in Vol I of his Werke.
185.8. EXTERNAL LINKS
185.7.5
583
Modern authors
• Cox, David A. (1989), Primes of the form x2 + n y2 , New York: Wiley, ISBN 0-471-50654-0 • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X • Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, doi:10.1007/9783-662-12893-0, ISBN 3-540-66957-4
185.8 External links • Weisstein, Eric W., “Biquadratic Reciprocity Theorem”, MathWorld. These two papers by Franz Lemmermeyer contain proofs of Burde’s law and related results: • Rational Quartic Reciprocity • Rational Quartic Reciprocity II
Chapter 186
Quasi-algebraically closed field In mathematics, a field F is called quasi-algebraically closed (or C1 ) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasialgebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper (Tsen 1936); and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper (Lang 1952). The idea itself is attributed to Lang’s advisor Emil Artin. Formally, if P is a non-constant homogeneous polynomial in variables X1 , ..., XN, and of degree d satisfying d
186.1 Examples • Any algebraically closed field is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero.[1] • Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.[2][3][4] • Algebraic function fields over algebraically closed fields are quasi-algebraically closed by Tsen’s theorem.[3][5] • The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.[3] • A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.[3][6] • A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.[7]
186.2 Properties • Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed. • The Brauer group of a finite extension of a quasi-algebraically closed field is trivial.[8][9][10] • A quasi-algebraically closed field has cohomological dimension at most 1.[10] 584
186.3. CK FIELDS
186.3
585
C fields
Quasi-algebraically closed fields are also called C 1 . A Ck field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided dk < N, for k ≥ 1.[11] The condition was first introduced and studied by Lang.[10] If a field is Cᵢ then so is a finite extension.[11][12] The C0 fields are precisely the algebraically closed fields.[13][14] Lang and Nagata proved that if a field is Ck, then any extension of transcendence degree n is Ck₊n.[15][16][17] The smallest k such that K is a C field ( ∞ if no such number exists), is called the diophantine dimension dd(K) of K.[13]
186.3.1 C 2 fields Every finite field is C2 .[7] Properties Suppose that the field k is C 2 . • Any skew field D finite over k as centre has the property that the reduced norm D∗ → k∗ is surjective.[16] • Every quadratic form in 5 or more variables over k is isotropic.[16] Artin’s conjecture Artin conjectured that p-adic fields were C 2 , but Guy Terjanian found p-adic counterexamples for all p.[18][19] The Ax–Kochen theorem applied methods from model theory to show that Artin’s conjecture was true for Qp with p large enough (depending on d).
186.3.2
Weakly Ck fields
A field K is weakly Ck,d if for every homogeneous polynomial of degree d in N variables satisfying dk < N the Zariski closed set V(f) of Pn (K) contains a subvariety which is Zariski closed over K. A field which is weakly Ck,d for every d is weakly Ck.[2] Properties • A Ck field is weakly Ck.[2] • A perfect PAC weakly Ck field is Ck.[2] • A field K is weakly Ck,d if and only if every form satisfying the conditions has a point x defined over a field which is a primary extension of K.[20] • If a field is weakly Ck, then any extension of transcendence degree n is weakly Ck₊n.[17] • Any extension of an algebraically closed field is weakly C1 .[21] • Any field with procyclic absolute Galois group is weakly C1 .[21] • Any field of positive characteristic is weakly C2 .[21] • If the field of rational numbers is weakly C1 , then every field is weakly C1 .[21]
586
CHAPTER 186. QUASI-ALGEBRAICALLY CLOSED FIELD
186.4 See also • Brauer’s theorem on forms • Tsen rank
186.5 Citations [1] Fried & Jarden (2008) p.455 [2] Fried & Jarden (2008) p.456 [3] Serre (1979) p.162 [4] Gille & Szamuley (2006) p.142 [5] Gille & Szamuley (2006) p.143 [6] Gille & Szamuley (2006) p.144 [7] Fried & Jarden (2008) p.462 [8] Lorenz (2008) p.181 [9] Serre (1979) p.161 [10] Gille & Szamuely (2006) p.141 [11] Serre (1997) p.87 [12] Lang (1997) p.245 [13] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 3-540-37888-X. [14] Lorenz (2008) p.116 [15] Lorenz (2008) p.119 [16] Serre (1997) p.88 [17] Fried & Jarden (2008) p.459 [18] Terjanian, Guy (1966). “Un contre-example à une conjecture d'Artin”. C. R. Acad. Sci. Paris Sér. A-B (in French) 262: A612. Zbl 0133.29705. [19] Lang (1997) p.247 [20] Fried & Jarden (2008) p.457 [21] Fried & Jarden (2008) p.461
186.6 References • Ax, James; Kochen, Simon (1965). “Amer. J. Math.” 87. pp. 605–630. Zbl 0136.32805. • Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. • Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001. • Greenberg, M.J. (1969). Lectures of forms in many variables. Mathematics Lecture Note Series. New YorkAmsterdam: W.A. Benjamin. Zbl 0185.08304.
186.6. REFERENCES
587
• Lang, Serge (1952), “On quasi algebraic closure”, Annals of Mathematics 55: 373–390, doi:10.2307/1969785, Zbl 0046.26202 • Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051. • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 109–126. ISBN 978-0-387-72487-4. Zbl 1130.12001. • Serre, Jean-Pierre (1979). Local fields. Graduate Texts in Mathematics 67. Translated from the French by Marvin Jay Greenberg. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016. • Serre, Jean-Pierre (1997). Galois cohomology. Springer-Verlag. ISBN 3-540-61990-9. Zbl 0902.12004. • Tsen, C. (1936), “Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper”, J. Chinese Math. Soc. 171: 81–92, Zbl 0015.38803
Chapter 187
Quasi-finite field In mathematics, a quasi-finite field[1] is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite (i.e. non-archimedean local fields), but the theory applies equally well when the residue field is only assumed quasi-finite.[2]
187.1 Formal definition A quasi-finite field is a perfect field K together with an isomorphism of topological groups
ˆ → Gal(Ks /K), ϕ:Z where Ks is an algebraic closure of K (necessarily separable because K is perfect). The field extension Ks/K is infinite, b is the profinite completion of integers with and the Galois group is accordingly given the Krull topology. The group Z respect to its subgroups of finite index. This definition is equivalent to saying that K has a unique (necessarily cyclic) extension Kn of degree n for each integer n ≥ 1, and that the union of these extensions is equal to Ks.[3] Moreover, as part of the structure of the quasi-finite field, there is a generator Fn for each Gal(Kn/K), and the generators must be coherent, in the sense that if n divides m, the restriction of Fm to Kn is equal to Fn.
187.2 Examples The most basic example, which motivates the definition, is the finite field K = GF(q). It has a unique cyclic extension of degree n, namely Kn = GF(qn ). The union of the Kn is the algebraic closure Ks. We take Fn to be the Frobenius element; that is, Fn(x) = xq . Another example is K = C((T)), the ring of formal Laurent series in T over the field C of complex numbers. (These are simply formal power series in which we also allow finitely many terms of negative degree.) Then K has a unique cyclic extension
Kn = C((T 1/n )) of degree n for each n ≥ 1, whose union is an algebraic closure of K called the field of Puiseux series, and that a generator of Gal(Kn/K) is given by
Fn (T 1/n ) = e2πi/n T 1/n . This construction works if C is replaced by any algebraically closed field C of characteristic zero.[4] 588
187.3. NOTES
589
187.3 Notes [1] (Artin & Tate 2009, §XI.3) say that the field satisfies “Moriya’s axiom” [2] As shown by Mikao Moriya (Serre 1979, chapter XIII, p. 188) [3] (Serre 1979, §XIII.2 exercise 1, p. 192) [4] (Serre 1979, §XIII.2, p. 191)
187.4 References • Artin, Emil; Tate, John (2009) [1967], Class field theory, American Mathematical Society, ISBN 978-0-82184426-7, MR 2467155, Zbl 1179.11040 • Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Translated from the French by Marvin Jay Greenberg, Springer-Verlag, ISBN 0-387-90424-7, MR 554237, Zbl 0423.12016
Chapter 188
Quaternionic structure In mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field. A quaternionic structure is a triple (G,Q,q) where G is an elementary abelian group of exponent 2 with a distinguished element −1, Q is a pointed set with distinguished element 1, and q is a symmetric surjection G×G → Q satisfying axioms
1.
q(a, (−1)a) = 1,
2. 3.
q(a, b) = q(a, c) ⇔ q(a, bc) = 1, . q(a, b) = q(c, d) ⇒ ∃x|q(a, b) = q(a, x), q(c, d) = q(c, x)
Every field F gives rise to a Q-structure by taking G to be F ∗ /F ∗2 , Q the set of Brauer classes of quaternion algebras in the Brauer group of F with the split quaternion algebra as distinguished element and q(a,b) the quaternion algebra (a,b)F.
188.1 References • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
590
Chapter 189
Ramification (mathematics) For other uses, see Ramification. In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be
Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y.
seen to have two branches differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing together of the fibers of the mapping.
189.1 In complex analysis In complex analysis, the basic model can be taken as the z → zn mapping in the complex plane, near z = 0. This is the standard local picture in Riemann surface theory, of ramification of order n. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. See also branch point.
189.2 In algebraic topology In a covering map the Euler-Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The z → zn mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler-Poincaré characteristic 0), but with the whole disk the Euler-Poincaré characteristic is 1, n – 1 being the 'lost' points as the n sheets come together at z = 0. In geometric terms, ramification is something that happens in codimension two (like knot theory, and monodromy); since real codimension two is complex codimension one, the local complex example sets the pattern for higher591
592
CHAPTER 189. RAMIFICATION (MATHEMATICS)
dimensional complex manifolds. In complex analysis, sheets can't simply fold over along a line (one variable), or codimension one subspace in the general case. The ramification set (branch locus on the base, double point set above) will be two real dimensions lower than the ambient manifold, and so will not separate it into two 'sides’, locally―there will be paths that trace round the branch locus, just as in the example. In algebraic geometry over any field, by analogy, it also happens in algebraic codimension one.
189.3 In algebraic number theory 189.3.1
In algebraic extensions of Q
See also Splitting of prime ideals in Galois extensions Ramification in algebraic number theory means prime numbers factoring into some repeated prime ideal factors. Let R be the ring of integers of an algebraic number field K and P a prime ideal of R. For each extension field L of K we can consider the integral closure S of R in L and the ideal PS of S. This may or may not be prime, but assuming [L:K] is finite it is a product of prime ideals P 1 e(1) ... Pke(k) where the Pi are distinct prime ideals of S. Then P is said to ramify in L if e(i) > 1 for some i. If for all i e(i) = 1 it is said to be unramified. In other words, P ramifies in L if the ramification index e(i) is greater than one for some Pi. An equivalent condition is that S/PS has a non-zero nilpotent element: it is not a product of finite fields. The analogy with the Riemann surface case was already pointed out by Richard Dedekind and Heinrich M. Weber in the nineteenth century. The ramification is encoded in K by the relative discriminant and in L by the relative different. The former is an ideal of the ring of integers of K and is divisible by P if and only if some ideal Pi of S dividing P is ramified. The latter is an ideal of the ring of integers of L and is divisible by the prime ideal Pi of S precisely when Pi is ramified. The ramification is tame when the ramification indices e(i) are all relatively prime to the residue characteristic p of P, otherwise wild. This condition is important in Galois module theory. A finite generically étale extension B/A of Dedekind domains is tame iff the trace Tr : B → A is surjective.
189.3.2
In local fields
Main article: Ramification of local fields The more detailed analysis of ramification in number fields can be carried out using extensions of the p-adic numbers, because it is a local question. In that case a quantitative measure of ramification is defined for Galois extensions, basically by asking how far the Galois group moves field elements with respect to the metric. A sequence of ramification groups is defined, reifying (amongst other things) wild (non-tame) ramification. This goes beyond the geometric analogue.
189.4 In algebra Main article: Ramification theory of valuations In valuation theory, the ramification theory of valuations studies the set of extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.
189.5 In algebraic geometry There is also corresponding notion of unramified morphism in algebraic geometry. It serves to define étale morphisms.
189.6. SEE ALSO
593
Let f : X → Y be a morphism of schemes. The support sheaf ΩX/Y is called the ramification ( of the quasicoherent ) locus of f and the image of the ramification locus, f SuppΩX/Y , is called the branch locus of f . If ΩX/Y = 0 we say that f is formally unramified and if f is also of locally finite presentation we say that f is unramified [see Vakil’s notes].
189.6 See also • Eisenstein polynomial • Newton polygon • Puiseux expansion • Branched covering
189.7 References • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Vakil, Ravi, “Foundations of Algebraic Geometry”, Lecture Notes, http://math.stanford.edu/~{}vakil/216blog/
189.8 External links • Ramification in number fields at PlanetMath.org.
Chapter 190
Ramification group In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.
190.1 Ramification groups in lower numbering Ramification groups are a refinement of the Galois group G of a finite L/K Galois extension of local fields. We shall write w, OL , p for the valuation, the ring of integers and its maximal ideal for L . As a consequence of Hensel’s lemma, one can write OL = OK [α] for some α ∈ L where OK is the ring of integers of K .[1] (This is stronger than the primitive element theorem.) Then, for each integer i ≥ −1 , we define Gi to be the set of all s ∈ G that satisfies the following equivalent conditions. • (i) s operates trivially on OL /pi+1 . • (ii) w(s(x) − x) ≥ i + 1 for all x ∈ OL • (iii) w(s(α) − α) ≥ i + 1. The group Gi is called i -th ramification group. They form a decreasing filtration,
G−1 = G ⊃ G0 ⊃ G1 ⊃ . . . {∗}. In fact, the Gi are normal by (i) and trivial for sufficiently large i by (iii). For the lowest indices, it is customary to call G0 the inertia subgroup of G because of its relation to splitting of prime ideals, while G1 the wild inertia subgroup of G . The quotient G0 /G1 is called the tame quotient. The Galois group G and its subgroups Gi are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular, • G/G0 = Gal(l/k), where l, k are the (finite) residue fields of L, K .[2] • G0 = 1 ⇔ L/K is unramified. • G1 = 1 ⇔ L/K is tamely ramified (i.e., the ramification index is prime to the residue characteristic.) The study of ramification groups reduces to the totally ramified case since one has Gi = (G0 )i for i ≥ 0 . One also defines the function iG (s) = w(s(α) − α), s ∈ G . (ii) in the above shows iG is independent of choice of α and, moreover, the study of the filtration Gi is essentially equivalent to that of iG .[3] iG satisfies the following: for s, t ∈ G , • iG (s) ≥ i + 1 ⇔ s ∈ Gi . 594
190.2. RAMIFICATION GROUPS IN UPPER NUMBERING
595
• iG (tst−1 ) = iG (s). • iG (st) ≥ min{iG (s), iG (t)}. Fix a uniformizer π of L . Then s 7→ s(π)/π induces the injection Gi /Gi+1 → UL,i /UL,i+1 , i ≥ 0 where × UL,0 = OL , UL,i = 1 + pi . (The map actually does not depend on the choice of the uniformizer.[4] ) It follows from [5] this • G0 /G1 is cyclic of order prime to p • Gi /Gi+1 is a product of cyclic groups of order p . In particular, G1 is a p-group and G is solvable. The ramification groups can be used to compute the different DL/K of the extension L/K and that of subextensions:[6]
w(DL/K ) =
∑
iG (s) =
∞ ∑ (|Gi | − 1). 0
s̸=1
If H is a normal subgroup of G , then, for σ ∈ G , iG/H (σ) =
1 eL/K
∑
s7→σ iG (s)
.[7]
Combining this with the above one obtains: for a subextension F /K corresponding to H ,
vF (DF /K ) =
1 eL/F
∑
iG (s).
s̸∈H
−1 −1 [8] If s ∈ Gi , t ∈ Gj , i, j ≥ 1 , then ∑sts t ∈ Gi+j+1 . In the terminology of Lazard, this can be understood to mean the Lie algebra gr(G1 ) = i≥1 Gi /Gi+1 is abelian.
190.1.1
Example
Let K be generated by x1 =
√ √ √ √ 2 + 2 . The conjugates of x1 are x2 = 2 − 2 , x3 = - x1 , x4 = - x2 .
A little √ computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it π. 2 generates π2 ; (2)=π4 . Now x1 -x3 =2x1 , which is in π5 . √ √ and x1 -x2 = 4 − 2 2 , which is in π3 . Various methods show that the Galois group of K is C4 , cyclic of order 4. Also: G0 = G1 = G2 = C4 . and G3 = G4 =(13)(24). w(DK/Q ) = 3+3+3+1+1 = 11. so that the different DK/Q =π11 . x1 satisfies x4 −4x2 +2, which has discriminant 2048=211 .
190.2 Ramification groups in upper numbering If u is a real number ≥ −1 , let Gu denote Gi where i the least integer ≥ u . In other words, s ∈ Gu ⇔ iG (s) ≥ u+1. Define ϕ by[9] ∫
u
ϕ(u) = 0
dt (G0 : Gt )
596
CHAPTER 190. RAMIFICATION GROUP
where, by convention, (G0 : Gt ) is equal to (G−1 : G0 )−1 if t = −1 and is equal to 1 for −1 < t ≤ 0 .[10] Then ϕ(u) = u for −1 ≤ u ≤ 0 . It is immediate that ϕ is continuous and strictly increasing, and thus has the continuous inverse function ψ defined on [−1, ∞) . Define Gv = Gψ(v) . Gv is then called the v-th ramification group in upper numbering. In other words, Gϕ(u) = Gu . Note G−1 = G, G0 = G0 . The upper numbering is defined so as to be compatible with passage to quotients:[11] if H is normal in G , then (G/H)v = Gv H/H for all v (whereas lower numbering is compatible with passage to subgroups.) Herbrand’s theorem states that the ramification groups in the lower numbering satisfy Gu H/H = (G/H)v (for v = ϕL/F (u) where L/F is the subextension corresponding to H ), and that the ramification groups in the upper numbering satisfy Gu H/H = (G/H)u .[12][13] This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the absolute Galois group of a local field) from the inverse system of ramification groups for finite subextensions. The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if G is abelian, then the jumps in the filtration Gv are integers; i.e., Gi = Gi+1 whenever ϕ(i) is not an integer.[14] The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of Gn (L/K) under the isomorphism
G(L/K)ab ↔ K ∗ /NL/K (L∗ ) is just[15]
n n ∩ NL/K (L∗ )) . /(UK UK
190.3 Notes [1] Neukirch (1999) p.178 [2] since G/G0 is canonically isomorphic to the decomposition group. [3] Serre (1979) p.62 [4] Conrad [5] Use UL,0 /UL,1 ≃ l× and UL,i /UL,i+1 ≈ l+ [6] Serre (1979) 4.1 Prop.4, p.64 [7] Serre (1979) 4.1. Prop.3, p.63 [8] Serre (1979) 4.2. Proposition 10. [9] Serre (1967) p.156 [10] Neukirch (1999) p.179 [11] Serre (1967) p.155 [12] Neukirch (1999) p.180 [13] Serre (1979) p.75 [14] Neukirch (1999) p.355 [15] Snaith (1994) pp.30-31
190.4. SEE ALSO
597
190.4 See also • Ramification theory of valuations
190.5 References • B. Conrad, Math 248A. Higher ramification groups • Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001. • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Serre, Jean-Pierre (1967). “VI. Local class field theory”. In Cassels, J.W.S.; Fröhlich, A. Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. London: Academic Press. pp. 128–161. Zbl 0153.07403. • Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics 67. Translated from the French by Marvin Jay Greenberg. Berlin, New York: Springer-Verlag. ISBN 0-387-90424-7. MR 0554237. Zbl 0423.12016. • Snaith, Victor P. (1994). Galois module structure. Fields Institute monographs. Providence, RI: American Mathematical Society. ISBN 0-8218-0264-X. Zbl 0830.11042.
Chapter 191
Ramification theory of valuations In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.
191.1 Galois case The structure of the set of extensions is known better when L/K is Galois.
191.1.1
Decomposition group and inertia group
Let (K, v) be a valued field and let L be a finite Galois extension of K. Let Sv be the set of equivalence classes of extensions of v to L and let G be the Galois group of L over K. Then G acts on Sv by σ[w] = [w ○ σ] (i.e. w is a representative of the equivalence class [w] ∈ Sv and [w] is sent to the equivalence class of the composition of w with the automorphism σ : L → L; this is independent of the choice of w in [w]). In fact, this action is transitive. Given a fixed extension w of v to L, the decomposition group of w is the stabilizer subgroup Gw of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ Sv. Let mw denote the maximal ideal of w inside the valuation ring Rw of w. The inertia group of w is the subgroup Iw of Gw consisting of elements σ such that σx ≡ x (mod mw) for all x in Rw. In other words, Iw consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of Gw. The reduced ramification index e(w/v) is independent of w and is denoted e(v). Similarly, the relative degree f(w/v) is also independent of w and is denoted f(v).
191.2 See also • ramification group
191.3 References • Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001. • Zariski, Oscar; Samuel, Pierre (1976) [1960]. Commutative algebra, Volume II. Graduate Texts in Mathematics 29. New York, Heidelberg: Springer-Verlag. Chapter VI. ISBN 978-0-387-90171-8. Zbl 0322.13001.
598
Chapter 192
Rational number “Rationals” redirects here. For other uses, see Rational (disambiguation). In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, p and q, with the denominator q not equal to zero.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold Q , Unicode ℚ);[2] it was thus denoted in 1895 by Peano after quoziente, Italian for "quotient". The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal). A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.[1] The rational numbers can be formally defined as the equivalence classes of the quotient set (Z × (Z \ {0})) / ~, where the cartesian product Z × (Z \ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n is not 0 (n ≠ 0), and "~" is the equivalence relation defined by (m1 ,n1 ) ~ (m2 ,n2 ) if, and only if, m1 n2 − m2 n1 = 0. In abstract algebra, the rational numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, being the field of fractions for the ring of integers. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.[3] In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals. Zero divided by any other integer equals zero; therefore, zero is a rational number (but division by zero is undefined).
192.1 Terminology The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, the adjective rational often means that the underlying field considered is the field Q of rational numbers. Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a “polynomial over the rationals”. However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.
192.2 Arithmetic See also: Fraction (mathematics) § Arithmetic with fractions
599
600
CHAPTER 192. RATIONAL NUMBER
192.2.1
Embedding of integers
Any integer n can be expressed as the rational number n/1.
192.2.2 a b
=
192.2.3
Equality c d
if and only if ad = bc.
Ordering
Where both denominators are positive: a b
<
c d
if and only if ad < bc.
If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations: −a a = −b b and a −a = . −b b
192.2.4
Addition
Two fractions are added as follows: a c ad + bc + = . b d bd
192.2.5
Subtraction
a c ad − bc − = . b d bd
192.2.6
Multiplication
The rule for multiplication is: ac a c · = . b d bd
192.2.7
Division
Where c ≠ 0: c ad a ÷ = . b d bc Note that division is equivalent to multiplying by the reciprocal of the divisor fraction: a d ad = × . bc b c
192.3. CONTINUED FRACTION REPRESENTATION
192.2.8
601
Inverse
Additive and multiplicative inverses exist in the rational numbers:
−
(a) b
=
192.2.9
−a a = b −b
and
( a )−1 b
=
b if a ̸= 0. a
Exponentiation to integer power
If n is a non-negative integer, then ( a )n b
=
an bn
and (if a ≠ 0): ( a )−n b
=
bn . an
192.3 Continued fraction representation Main article: Continued fraction A finite continued fraction is an expression such as
1
a0 +
,
1
a1 + a2 +
1 ..
.+
1 an
where an are integers. Every rational number a/b has two closely related expressions as a finite continued fraction, whose coefficients an can be determined by applying the Euclidean algorithm to (a,b).
192.4 Formal construction Mathematically we may construct the rational numbers as equivalence classes of ordered pairs of integers (m,n), with n ≠ 0. This space of equivalence classes is the quotient space (Z × (Z \ {0})) / ~, where (m1 ,n1 ) ~ (m2 ,n2 ) if, and only if, m1 n2 − m2 n1 = 0. We can define addition and multiplication of these pairs with the following rules:
(m1 , n1 ) + (m2 , n2 ) ≡ (m1 n2 + n1 m2 , n1 n2 ) (m1 , n1 ) × (m2 , n2 ) ≡ (m1 m2 , n1 n2 ) and, if m2 ≠ 0, division by (m1 , n1 ) ≡ (m1 n2 , n1 m2 ) . (m2 , n2 ) The equivalence relation (m1 ,n1 ) ~ (m2 ,n2 ) if, and only if, m1 n2 − m2 n1 = 0 is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set (Z × (Z \
602
CHAPTER 192. RATIONAL NUMBER
A diagram showing a representation of the equivalent classes of pairs of integers
{0})) / ~, i.e. we identify two pairs (m1 ,n1 ) and (m2 ,n2 ) if they are equivalent in the above sense. (This construction can be carried out in any integral domain: see field of fractions.) We denote by [(m1 ,n1 )] the equivalence class containing (m1 ,n1 ). If (m1 ,n1 ) ~ (m2 ,n2 ) then, by definition, (m1 ,n1 ) belongs to [(m2 ,n2 )] and (m2 ,n2 ) belongs to [(m1 ,n1 )]; in this case we can write [(m1 ,n1 )] = [(m2 ,n2 )]. Given any equivalence class [(m,n)] there are a countably infinite number of representation, since
· · · = [(−2m, −2n)] = [(−m, −n)] = [(m, n)] = [(2m, 2n)] = · · · . The canonical choice for [(m,n)] is chosen so that n is positive and gcd(m,n) = 1, i.e. m and n share no common factors, i.e. m and n are coprime. For example, we would write [(1,2)] instead of [(2,4)] or [(−12,−24)], even though [(1,2)] = [(2,4)] = [(−12,−24)]. We can also define a total order on Q. Let ∧ be the and-symbol and ∨ be the or-symbol. We say that [(m1 ,n1 )] ≤ [(m2 ,n2 )] if:
(n1 n2 > 0 ∧ m1 n2 ≤ n1 m2 ) ∨ (n1 n2 < 0 ∧ m1 n2 ≥ n1 m2 ). The integers may be considered to be rational numbers by the embedding that maps m to [(m,1)].
192.5. PROPERTIES
603
192.5 Properties
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8
...
2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8
...
3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8
...
4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8
...
5/1 5/2 5/3 5/4 5/5 5/6 5/7 5/8
...
6/1 6/2 6/3 6/4 6/5 6/6 6/7 6/8
...
7/1 7/2 7/3 7/4 7/5 7/6 7/7 7/8
...
8/1 8/2 8/3 8/4 8/5 8/6 8/7 8/8
...
...
...
...
...
...
...
...
...
...
A diagram illustrating the countability of the rationals
The set Q, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers Z. The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of Q. The rational numbers are therefore the prime field for characteristic zero. The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the algebraic numbers. The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set. The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that
c a < b d (where b, d are positive), we have
604
CHAPTER 192. RATIONAL NUMBER
a ad + bc c < < . b 2bd d Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.
192.6 Real numbers and topological properties The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space by using the absolute difference metric d(x,y) = |x − y|, and this yields a third topology on Q. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of Q under the metric d(x,y) = |x − y|, above.
192.7
p-adic numbers
See also: p-adic Number In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field: Let p be a prime number and for any non-zero integer a, let |a|p = p−n , where pn is the highest power of p dividing a. In addition set |0|p = 0. For any rational number a/b, we set |a/b|p = |a|p / |b|p. Then dp(x,y) = |x − y|p defines a metric on Q. The metric space (Q,dp) is not complete, and its completion is the p-adic number field Qp. Ostrowski’s theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.
192.8 See also • Floating point • Ford circles • Niven’s theorem • Rational data type
192.9 References [1] Rosen, Kenneth (2007). Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105,158– 160. ISBN 978-0-07-288008-3. [2] Rouse, Margaret. “Mathematical Symbols”. Retrieved 1 April 2015. [3] Gilbert, Jimmie; Linda, Gilbert (2005). Elements of Modern Algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244. ISBN 0-534-40264-X.
192.10. EXTERNAL LINKS
605
192.10 External links • Hazewinkel, Michiel, ed. (2001), “Rational number”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4 • “Rational Number” From MathWorld – A Wolfram Web Resource
Chapter 193
Rational reciprocity law In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity. As an example, there are rational biquadratic and octic reciprocity laws. Define the symbol (x|p)k to be +1 if x is a k-th power modulo the prime p and −1 otherwise. Let p and q be distinct primes congruent to 1 modulo 4, such that (p|q)2 = (q|p)2 = +1. Let p = a2 + b2 and q = A2 + B2 with aA odd. Then
(p|q)4 (q|p)4 = (−1)(q−1)/4 (aB − bA|q)2 . If in addition p and q are congruent to 1 modulo 8, let p = c2 + 2d2 and q = C 2 + 2D2 . Then
(p|q)8 = (q|p)8 = (aB − bA|q)4 (cD − dC|q)2 .
193.1 References • Burde, K. (1969), “Ein rationales biquadratisches Reziprozitätsgesetz”, J. Reine Angew. Math. (in German) 235: 175–184, Zbl 0169.36902 • Lehmer, Emma (1978), “Rational reciprocity laws”, The American Mathematical Monthly 85 (6): 467–472, doi:10.2307/2320065, ISSN 0002-9890, MR 0498352, Zbl 0383.10003 • Lemmermeyer, Franz (2000), Reciprocity laws. From Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer-Verlag, pp. 153–183, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002 • Williams, Kenneth S. (1976), “A rational octic reciprocity law”, Pacific Journal of Mathematics 63 (2): 563– 570, doi:10.2140/pjm.1976.63.563, ISSN 0030-8730, MR 0414467, Zbl 0311.10004
606
Chapter 194
Rational variety In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to
K(U1 , . . . , Ud ), the field of all rational functions for some set {U1 , . . . , Ud } of indeterminates, where d is the dimension of the variety.
194.1 Rationality and parameterization Let V be an affine algebraic variety of dimension d defined by a prime ideal I=⟨f 1 , ..., fk⟩ in K[X1 , . . . , Xn ] . If V is rational, then there are n+1 polynomials g0 , ..., gn in K(U1 , . . . , Ud ) such that fi (g1 /g0 , . . . , gn /g0 ) = 0. In order words, we have a rational parameterization xi = gg0i (u1 , . . . , ud ) of the variety. Conversely, such a rational parameterization induces a field homomorphism of the field of functions of V into K(U1 , . . . , Ud ) . But this homomorphism is not necessarily onto. If such a parameterization exists, the variety is said unirational. Lüroth’s theorem (see below) implies that unirational curves are rational. Castelnuovo’s theorem implies also that, in characteristic zero, every unirational surface is rational.
194.2 Rationality questions A rationality question asks whether a given field extension is rational, in the sense of being (up to isomorphism) the function field of a rational variety; such field extensions are also described as purely transcendental. More precisely, the rationality question for the field extension K ⊂ L is this: is L isomorphic to a rational function field over K in the number of indeterminates given by the transcendence degree? There are several different variations of this question, arising from the way in which the fields K and L are constructed. For example, let K be a field, and let
{y1 , . . . , yn } be indeterminates over K and let L be the field generated over K by them. Consider a finite group G permuting those indeterminates over K. By standard Galois theory, the set of fixed points of this group action is a subfield of L , typically denoted LG . The rationality question for K ⊂ LG is called Noether’s problem and asks if this field of fixed points is or is not a purely transcendental extension of K. In the paper (Noether 1918) on Galois theory she studied the problem of parameterizing the equations with given Galois group, which she reduced to “Noether’s problem”. (She first mentioned this problem in (Noether 1913) where she attributed the problem to E. Fischer.) She showed this was true for n = 2, 3, or 4. R. G. Swan (1969) found a counter-example to the Noether’s problem, with n = 47 and G a cyclic group of order 47. 607
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CHAPTER 194. RATIONAL VARIETY
194.3 Lüroth’s theorem Main article: Lüroth’s theorem A celebrated case is Lüroth’s problem, which Jacob Lüroth solved in the nineteenth century. Lüroth’s problem concerns subextensions L of K(X), the rational functions in the single indeterminate X. Any such field is either equal to K or is also rational, i.e. L = K(F) for some rational function F. In geometrical terms this states that a non-constant rational map from the projective line to a curve C can only occur when C also has genus 0. That fact can be read off geometrically from the Riemann–Hurwitz formula. Even though Lüroth’s theorem is often thought as a non elementary result, several elementary short proofs have been discovered for long. These simple proofs use only the basics of field theory and Gauss’s lemma for primitive polynomials (see e.g. [1] ).
194.4 Unirationality A unirational variety V over a field K is one dominated by a rational variety, so that its function field K(V) lies in a pure transcendental field of finite type (which can be chosen to be of finite degree over K(V) if K is infinite). The solution of Lüroth’s problem shows that for algebraic curves, rational and unirational are the same, and Castelnuovo’s theorem implies that for complex surfaces unirational implies rational, because both are characterized by the vanishing of both the arithmetic genus and the second plurigenus. Zariski found some examples (Zariski surfaces) in characteristic p > 0 that are unirational but not rational. Clemens & Griffiths (1972) showed that a cubic three-fold is in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. Their work used an intermediate Jacobian. Iskovskih & Manin (1971) showed that all non-singular quartic threefolds are irrational, though some of them are unirational. Artin & Mumford (1972) found some unirational 3-folds with non-trivial torsion in their third cohomology group, which implies that they are not rational. For any field K, János Kollár proved in 2000 that a smooth cubic hypersurface of dimension at least 2 is unirational if it has a point defined over K. This is an improvement of many classical results, beginning with the case of cubic surfaces (which are rational varieties over an algebraic closure). Other examples of varieties that are shown to be unirational are many cases of the moduli space of curves.[2]
194.5 Rationally connected variety A rationally connected variety V is a projective algebraic variety over an algebraically closed field such that through every two points there passes the image of a regular map from the projective line into V. Equivalently, a variety is rationally connected if every two points are connected by a rational curve contained in the variety.[3] This definition differs form that of path connectedness only by the nature of the path, but is very different, as the only algebraic curves which are rationally connected are the rational ones. Every rational variety, including the projective spaces, is rationally connected, but the converse is false. The class of the rationally connected varieties is thus a generalization of the class of the rational varieties. Unirational varieties are rationally connected, but it is not known if the converse holds.
194.6 See also • Rational curve • Rational surface • Severi–Brauer variety • Birational geometry
194.7. NOTES
609
194.7 Notes [1] Bensimhoun, Michael (May 2004). “Another elementary proof of Luroth’s theorem” (PDF). Jerusalem. [2] János Kollár (2002). “Unirationality of cubic hypersurfaces”. Journal of the Institute of Mathematics of Jussieu 1 (3): 467–476. doi:10.1017/S1474748002000117. MR 1956057. [3] Kollár, János (1996), Rational Curves on Algebraic Varieties, Berlin, New York: Springer-Verlag.
194.8 References • Artin, Michael; Mumford, David (1972), “Some elementary examples of unirational varieties which are not rational”, Proceedings of the London Mathematical Society. Third Series 25: 75–95, doi:10.1112/plms/s325.1.75, ISSN 0024-6115, MR 0321934 • Clemens, C. Herbert; Griffiths, Phillip A. (1972), “The intermediate Jacobian of the cubic threefold”, Annals of Mathematics. Second Series (The Annals of Mathematics, Vol. 95, No. 2) 95 (2): 281–356, doi:10.2307/1970801, ISSN 0003-486X, JSTOR 1970801, MR 0302652 • Iskovskih, V. A.; Manin, Ju. I. (1971), “Three-dimensional quartics and counterexamples to the Lüroth problem”, Matematicheskii Sbornik, Novaya Seriya 86: 140–166, doi:10.1070/SM1971v015n01ABEH001536, MR 0291172 • Kollár, János; Smith, Karen E.; Corti, Alessio (2004), Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics 92, Cambridge University Press, ISBN 978-0-521-83207-6, MR 2062787 • Noether, Emmy (1913), “Rationale Funkionenkorper”, J. Ber. D. DMV 22: 316–319. • Noether, Emmy (1918), “Gleichungen mit vorgeschriebener Gruppe”, Mathematische Annalen 78: 221–229, doi:10.1007/BF01457099. • Swan, R. G. (1969), “Invariant rational functions and a problem of Steenrod”, Inventiones Mathematicae 7 (2): 148–158, doi:10.1007/BF01389798 • Martinet, J. (1971), “Exp. 372 Un contre-exemple à une conjecture d'E. Noether (d'après R. Swan);", Séminaire Bourbaki. Vol. 1969/70: Exposés 364–381, Lecture Notes in Mathematics 189, Berlin, New York: SpringerVerlag, MR 0272580
Chapter 195
Ray class field In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes. Every finite abelian extension of a number field is contained in one of its ray class fields. The term “ray class group” is a translation of the German term “Strahlklassengruppe”. Here “Strahl” is the German for a ray, and often means the positive real line, which appears in the positivity conditions defining ray class groups. Hasse (1926, p.6) uses “Strahl” to mean a certain group of ideals defined using positivity conditions, and uses “Strahlklasse” to mean a coset of this group. There are two slightly different notions of what a ray class field is, as authors differ in how the infinite primes are treated.
195.1 History Weber introduced ray class groups in 1897. Takagi proved the existence of the corresponding ray class fields in about 1920. Chevalley reformuated the definition of ray class groups in terms of ideals in 1933.
195.2 Ray class fields using ideals If m is an ideal of the ring of integers of a number field K and S is a subset of the real places, then the ray class group of m and S is the quotient group
I m /P m where I m is the group of fractional ideals co-prime to m, and the “ray” P m is the group of principal ideals generated by elements a with a ≡ 1 mod m that are positive at the places of S. When S consists of all real places, so that a is restricted to be totally positive, the group is called the narrow ray class group of m. Some authors use the term “ray class group” to mean “narrow ray class group”. A ray class field of K is the abelian extension of K associated to a ray class group by class field theory, and its Galois group is isomorphic to the corresponding ray class group. The proof of existence of a ray class field of a given ray class group is long and indirect and there is in general no known easy way to construct it (though explicit constructions are known in some special cases such as imaginary quadratic fields).
195.3 Ray class fields using ideles Chevalley redefined the ray class group of an ideal m and a set S of real places as the quotient of the idele class group by image of the group 610
195.4. EXAMPLES
∏
611
Up
where Up is given by: • The nonzero complex numbers for a complex place p • The positive real numbers for a real place p in S, and all nonzero real numbers for p not in S • The units of Kp for a finite place p not dividing m • The units of Kp congruent to 1 mod pn if pn is the maximal power of p dividing m. Some authors use a more general definition, where the group Up is allowed to be all nonzero real numbers for certain real places p. The ray class groups defined using ideles are naturally isomorphic to those defined using ideals. They are sometimes easier to handle theoretically because they are all quotients of a single group, and thus easier to compare. The ray class field of a ray class ∏group is the (unique) abelian extension L of K such that the norm of the idele class group CL of L is the image of Up in the idele class group of K.
195.4 Examples If K is the field of rational numbers and (m) is some non-zero ideal for an integer m, then the ray class group of (m) is isomorphic to the group of units of Z/mZ, and the ray class field is the field generated by the mth roots of unity. The Hilbert class field is the ray class field corresponding to the unit ideal and the empty set of real places, so it is the smallest ray class field. The narrow Hilbert class field is the ray class field corresponding to the unit ideal and the set of all real places, so it is the smallest narrow ray class field.
195.5 References • Hasse, Helmut (1926), “Bericht über neuere Unterschungen und Probleme aus der Theorie der algebraischen Zahlkörper.”, Jahresbericht der Deutschen Mathematiker-Vereinigung (Göttingen: Teubner) 35 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
Chapter 196
Real closed field “Artin–Schreier theorem” redirects here. For the branch of Galois theory, see Artin–Schreier theory. In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
196.1 Definitions A real closed field is a field F in which any of the following equivalent conditions are true: 1. F is elementarily equivalent to the real numbers. In other words it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in F if and only if it is true in the reals. (The choice of signature is not significant.) 2. There is a total order on F making it an ordered field such that, in this ordering, every positive element of F has a square root in F and any polynomial of odd degree with coefficients in F has at least one root in F. 3. F is a formally real field such that every polynomial of odd degree with coefficients in F has at least one root in F, and for every element a of F there is b in F such that a = b2 or a = −b2 . 4. F is not algebraically closed but its algebraic closure is a finite extension. √ 5. F is not algebraically closed but the field extension F ( −1) is algebraically closed. 6. There is an ordering on F which does not extend to an ordering on any proper algebraic extension of F. 7. F is a formally real field such that no proper algebraic extension of F is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.) 8. There is an ordering on F making it an ordered field such that, in this ordering, the intermediate value theorem holds for all polynomials over F with degree ≥ 0. 9. F is a real closed ring. If F is an ordered field, the Artin–Schreier theorem states that F has an algebraic extension, called the real closure K of F, such that K is a real closed field whose ordering is an extension of the given ordering on F, and is unique up to a unique isomorphism of fields identical on F [1] (note that every ring homomorphism between real closed fields automatically is order preserving, because x ≤ y if and only if ∃z y = x + z2 ). For example, the real closure of the ordered field of rational numbers is the field Ralg of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926. If (F,P) is an ordered field, and E is a Galois extension of F, then by Zorn’s Lemma there is a maximal ordered field extension (M,Q) with M a subfield of E containing F and the order on M extending P. M, together with its ordering Q, is called the relative real closure of (F,P) in E. We call (F,P) real closed relative to E if M is just F. When E is the algebraic closure of F the relative real closure of F in E is actually the real closure of F described earlier.[2] 612
196.2. MODEL THEORY: DECIDABILITY AND QUANTIFIER ELIMINATION
613
If F is a field (no ordering compatible with the field operations is assumed, nor is assumed that F is orderable) then F still has a real √closure, which may not be a field anymore, but just a real closed ring. For example, √ the real closure of the field √ Q( 2) is the ring Ralg × Ralg (the two copies correspond to the two orderings of Q( 2) ). On the other hand, if Q( 2) is considered as an ordered subfield of R , its real closure is again the field Ralg .
196.2 Model theory: decidability and quantifier elimination Although the theory of real closed fields was firstly developed by algebraists, it has received considerable attention from Model Theory. By adding to the ordered field axioms • an axiom asserting that every positive number has a square root, and • an axiom scheme asserting that all polynomials of odd degree have at least one root one obtains a first-order theory. Alfred Tarski (1951) proved that the theory of real closed fields in the first order language of partially ordered rings (consisting of the binary predicate symbols "=" and "≤", the operations of addition, subtraction and multiplication and the constant symbols 0,1) admits elimination of quantifiers. The most important model theoretic consequences hereof: The theory of real closed fields is complete, o-minimal and decidable. Decidability means that there exists at least one decision procedure, i.e., a well-defined algorithm for determining whether a sentence in the first order language of real closed fields is true. Euclidean geometry (without the ability to measure angles) is also a model of the real field axioms, and thus is also decidable. The decision procedures are not necessarily practical. The algorithmic complexities of all known decision procedures for real closed fields are very high, so that practical execution times can be prohibitively high except for very simple problems. The algorithm Tarski proposed for quantifier elimination has NONELEMENTARY complexity, meaning that no ·n ··
tower 22 can bound the execution time of the algorithm if n is the size of the problem. Davenport and Heintz (1988) proved that quantifier elimination is in fact (at least) doubly exponential: there exists a family Φ of formulas with n quantifiers, of length O(n) and constant degree such that any quantifier-free formula equivalent to Φ must Ω(n) Ω(n) involve polynomials of degree 22 and length 22 , using the Ω asymptotic notation. Ben-Or, Kozen, and Reif (1986) proved that the theory of real closed fields is decidable in exponential space, and therefore in doubly exponential time. Basu and Roy (1996) proved that there exists a well-behaved algorithm to decide the truth of a formula ∃x1 ,…,∃x P1 (x1 ,…,x )⋈0∧…∧P (x1 ,…,x )⋈0 where ⋈ is <, > or =, with complexity in arithmetic operations sk+1 dO(k) . In fact, the existential theory of the reals can be decided in PSPACE. Adding additional functions symbols, for example, the sine or the exponential function, can change the decidability of the theory. Yet another important model-theoretic property of real closed fields is that they are weakly o-minimal structures. Conversely, it is known that any weakly o-minimal ordered field must be real closed.[3]
196.3 Order properties A crucially important property of the real numbers is that it is an Archimedean field, meaning it has the Archimedean property that for any real number, there is an integer larger than it in absolute value. An equivalent statement is that for any real number, there are integers both larger and smaller. Such real closed fields that are not Archimedean, are non-Archimedean ordered fields. For example, any field of hyperreal numbers is real closed and non-Archimedean. The Archimedean property is related to the concept of cofinality. A set X contained in an ordered set F is cofinal in F if for every y in F there is an x in X such that y < x. In other words, X is an unbounded sequence in F. The cofinality of F is the size of the smallest cofinal set, which is to say, the size of the smallest cardinality giving an unbounded sequence. For example natural numbers are cofinal in the reals, and the cofinality of the reals is therefore ℵ0 . We have therefore the following invariants defining the nature of a real closed field F: • The cardinality of F.
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CHAPTER 196. REAL CLOSED FIELD
• The cofinality of F. To this we may add • The weight of F, which is the minimum size of a dense subset of F. These three cardinal numbers tell us much about the order properties of any real closed field, though it may be difficult to discover what they are, especially if we are not willing to invoke generalized continuum hypothesis. There are also particular properties which may or may not hold: • A field F is complete if there is no ordered field K properly containing F such that F is dense in K. If the cofinality of F is κ, this is equivalent to saying Cauchy sequences indexed by κ are convergent in F. • An ordered field F has the eta set property ηα, for the ordinal number α, if for any two subsets L and U of F of cardinality less than ℵα such that every element of L is less than every element of U, there is an element x in F with x larger than every element of L and smaller than every element of U. This is closely related to the model-theoretic property of being a saturated model; any two real closed fields are ηα if and only if they are ℵα -saturated, and moreover two ηα real closed fields both of cardinality ℵα are order isomorphic.
196.4 The generalized continuum hypothesis The characteristics of real closed fields become much simpler if we are willing to assume the generalized continuum hypothesis. If the continuum hypothesis holds, all real closed fields with cardinality the continuum and having the η1 property are order isomorphic. This unique field Ϝ can be defined by means of an ultrapower, as RN /M , where M is a maximal ideal not leading to a field order-isomorphic to R . This is the most commonly used hyperreal number field in non-standard analysis, and its uniqueness is equivalent to the continuum hypothesis. (Even without the continuum hypothesis we have that if the cardinality of the continuum is ℵβ then we have a unique ηᵦ field of size ηᵦ.) Moreover, we do not need ultrapowers to construct Ϝ, we can do so much more constructively as the subfield of series with a countable number of nonzero terms of the field R((G)) of formal power series on a totally ordered abelian divisible group G that is an η1 group of cardinality ℵ1 (Alling 1962). Ϝ however is not a complete field; if we take its completion, we end up with a field Κ of larger cardinality. Ϝ has the cardinality of the continuum which by hypothesis is ℵ1 , Κ has cardinality ℵ2 , and contains Ϝ as a dense subfield. It is not an ultrapower but it is a hyperreal field, and hence a suitable field for the usages of nonstandard analysis. It can be seen to be the higher-dimensional analogue of the real numbers; with cardinality ℵ2 instead of ℵ1 , cofinality ℵ1 instead of ℵ0 , and weight ℵ1 instead of ℵ0 , and with the η1 property in place of the η0 property (which merely means between any two real numbers we can find another).
196.5 Examples of real closed fields • the real algebraic numbers • the computable numbers • the definable numbers • the real numbers • superreal numbers • hyperreal numbers • the Puiseux series with real coefficients
196.6. NOTES
615
196.6 Notes [1] Rajwade (1993) pp. 222–223 [2] Efrat (2006) p. 177 [3] D. Macpherson et. al, (1998)
196.7 References • Alling, Norman L. (1962), “On the existence of real-closed fields that are ηα-sets of power ℵα.”, Trans. Amer. Math. Soc. 103: 341–352, doi:10.1090/S0002-9947-1962-0146089-X, MR 0146089 • Basu, Saugata, Richard Pollack, and Marie-Françoise Roy (2003) “Algorithms in real algebraic geometry” in Algorithms and computation in mathematics. Springer. ISBN 3-540-33098-4 (online version) • Michael Ben-Or, Dexter Kozen, and John Reif, The complexity of elementary algebra and geometry, Journal of Computer and Systems Sciences 32 (1986), no. 2, pp. 251–264. • Caviness, B F, and Jeremy R. Johnson, eds. (1998) Quantifier elimination and cylindrical algebraic decomposition. Springer. ISBN 3-211-82794-3 • Chen Chung Chang and Howard Jerome Keisler (1989) Model Theory. North-Holland. • Dales, H. G., and W. Hugh Woodin (1996) Super-Real Fields. Oxford Univ. Press. • Davenport, James H.; Heintz, Joos (1988). “Real quantifier elimination is doubly exponential”. J. Symb. Comput. 5 (1-2): 29–35. Zbl 0663.03015. • Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs 124. Providence, RI: American Mathematical Society. ISBN 0-8218-4041-X. Zbl 1103.12002. • Macpherson, D., Marker, D. and Steinhorn, C., Weakly o-minimal structures and real closed fields, Trans. of the American Math. Soc., Vol. 352, No. 12, 1998. • Mishra, Bhubaneswar (1997) "Computational Real Algebraic Geometry," in Handbook of Discrete and Computational Geometry. CRC Press. 2004 edition, p. 743. ISBN 1-58488-301-4 • Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022. • Alfred Tarski (1951) A Decision Method for Elementary Algebra and Geometry. Univ. of California Press. • Erdös, P.; Gillman, L.; Henriksen, M. (1955), “An isomorphism theorem for real-closed fields”, Ann. of Math. (2) 61: 542–554, MR 0069161
196.8 External links • Real Algebraic and Analytic Geometry Preprint Server • Model Theory preprint server
Chapter 197
Reciprocity law For the reciprocity law in photography, see Reciprocity (photography). In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.
197.1 Quadratic reciprocity Main article: quadratic reciprocity In terms of the Legendre symbol, the law of quadratic reciprocity for positive odd primes states ( )( ) p−1 q−1 p q = (−1) 2 2 . q p
197.2 Cubic reciprocity Main article: cubic reciprocity The law of cubic reciprocity for Eisenstein integers states that if α and β are primary (primes congruent to 2 mod 3) then ( ) ( ) α β = . β α 3
3
197.3 Quartic reciprocity Main article: quartic reciprocity
616
197.4. OCTIC RECIPROCITY
617
In terms of the quartic residue symbol, the law of quartic reciprocity for Gaussian integers states that if π and θ are primary (congruent to 1 mod (1+i)3 ) Gaussian primes then [ ][ ] −1 N π−1 N θ−1 θ π 4 . = (−1) 4 θ π
197.4 Octic reciprocity Main article: Octic reciprocity
197.5 Eisenstein reciprocity Main article: Eisenstein reciprocity Suppose that ζ is an l th root of unity for some odd prime l . The power character is the power of ζ such that ( ) N (p)−1 α ≡α l p l
(mod p)
for any prime ideal p of Z[ζ]. It is extended to other ideals by multiplicativity. The Eisenstein reciprocity law states that (a) α
= l
(α) a
l
for a any rational integer coprime to l and α any element of Z[ζ] that is coprime to a and l and congruent to a rational integer modulo (1–ζ)2 .
197.6 Kummer reciprocity Suppose that ζ is an lth root of unity for some odd regular prime l. Since l is regular, we can extend the symbol {} to ideals in a unique way such that { }n p q
{ =
pn q
}
where n is some integer prime to l such that pn is principal.
The Kummer reciprocity law states that { } { } p q = q p for p and q any distinct prime ideals of Z[ζ] other than (1–ζ).
197.7 Hilbert reciprocity Main article: Hilbert symbol In terms of the Hilbert symbol, Hilbert’s reciprocity law for an algebraic number field states that
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CHAPTER 197. RECIPROCITY LAW
∏ (a, b)v = 1 v
where the product is over all finite and infinite places. Over the rational numbers this is equivalent to the law of quadratic reciprocity. To see this take a and b to be distinct odd primes. Then Hilbert’s law becomes (p, q)∞ (p, q)2 (p, q)p (p, q)q = 1 But (p,q)p is equal to the Legendre symbol, (p,q)∞ is 1 if one of p and q is positive and –1 otherwise, and (p,q)2 is (–1)(p–1)(q–1)/4 . So for p and q positive odd primes Hilbert’s law is the law of quadratic reciprocity.
197.8 Artin reciprocity Main article: Artin reciprocity law In the language of ideles, the Artin reciprocity law for a finite extension L/K states that the Artin map from the idele class group CK to the abelianization Gal(L/K)ab of the Galois group vanishes on NL/K(CL), and induces an isomorphism
θ : CK /NL/K (CL ) → Gal(L/K)ab . Although it is not immediately obvious, the Artin reciprocity law easily implies all the previously discovered reciprocity laws, by applying it to suitable extensions L/K. For example, in the special case when K contains the nth roots of unity and L=K[a1/n ] is a Kummer extension of K, the fact that the Artin map vanishes on NL/K(CL) implies Hilbert’s reciprocity law for the Hilbert symbol.
197.9 Local reciprocity Hasse introduced a local analogue of the Artin reciprocity law, called the local reciprocity law. One form of it states that for a finite abelian extension of L/K of local fields, the Artin map is an isomorphism from K × /NL/K (L× ) onto the Galois group Gal(L/K) .
197.10 Explicit reciprocity laws Main article: Explicit reciprocity law In order to get a classical style reciprocity law from the Hilbert reciprocity law Π(a,b)p=1, one needs to know the values of (a,b)p for p dividing n. Explicit formulas for this are sometimes called explicit reciprocity laws.
197.11 Power reciprocity laws Main article: Power reciprocity law A power reciprocity law may be formulated as an analogue of the law of quadratic reciprocity in terms of the Hilbert symbols as[1] ( ) ( )−1 ∏ α β = (α, β)p . β n α n p|n∞
197.12. RATIONAL RECIPROCITY LAWS
619
197.12 Rational reciprocity laws Main article: Rational reciprocity law A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity.
197.13 Scholz’s reciprocity law Main article: Scholz’s reciprocity law
197.14 Shimura reciprocity Main article: Shimura’s reciprocity law
197.15 Weil reciprocity law Main article: Weil reciprocity law
197.16 Langlands reciprocity The Langlands program includes several conjectures for general reductive algebraic groups, which for the special of the group GL1 imply the Artin reciprocity law.
197.17 Yamamoto’s reciprocity law Main article: Yamamoto’s reciprocity law Yamamoto’s reciprocity law is a reciprocity law related to class numbers of quadratic number fields.
197.18 See also • Hilbert’s ninth problem • Stanley’s reciprocity theorem
197.19 References [1] Neukirch (1999) p.415
• Frei, Günther (1994), “The reciprocity law from Euler to Eisenstein”, in Chikara, Sasaki, The intersection of history and mathematics. Papers presented at the history of mathematics symposium, held in Tokyo, Japan, August 31 - September 1, 1990, Sci. Networks Hist. Stud. 15, Basel: Birkhäuser, pp. 67–90, MR 308080, Zbl 0818.01002
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• Hilbert, David (1897), “Die Theorie der algebraischen Zahlkörper”, Jahresbericht der Deutschen MathematikerVereinigung (in German) 4: 175–546, ISSN 0012-0456 • Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, ISBN 9783-540-62779-1, MR 1646901 • Lemmermeyer, Franz (2000), Reciprocity laws. From Euler to Eisenstein, Springer Monographs in Mathematics, Berlin: Springer-Verlag, ISBN 3-540-66957-4, MR 1761696, Zbl 0949.11002 • Lemmermeyer, Franz, Reciprocity laws. From Kummer to Hilbert • Neukirch, Jürgen (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften 322, Translated from the German by Norbert Schappacher, Berlin: Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021 • Stepanov, S. A. (2001), “Reciprocity laws”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Wyman, B. F. (1972), “What is a reciprocity law?", Amer. Math. Monthly 79: 571–586, JSTOR 2317083, MR 0308084 correction, ibid. 80 (1973), 281
Chapter 198
Regular extension In field theory, a branch of algebra, a field extension L/k is said to be regular if k is algebraically closed in L (i.e., k = kˆ where kˆ is the set of elements in L algebraic over k) and L is separable over k, or equivalently, L ⊗k k is an integral domain when k is the algebraic closure of k (that is, to say, L, k are linearly disjoint over k).[1][2]
198.1 Properties • Regularity is transitive: if F/E and E/K are regular then so is F/K.[3] • If F/K is regular then so is E/K for any E between F and K.[3] • The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.[2] • Any extension of an algebraically closed field is regular.[3][4] • An extension is regular if and only if it is separable and primary.[5] • A purely transcendental extension of a field is regular.
198.2 Self-regular extension There is also a similar notion: a field extension L/k is said to be self-regular if L ⊗k L is an integral domain. A self-regular extension is relatively algebraically closed in k.[6] However, a self-regular extension is not necessarily regular.
198.3 References [1] Fried & Jarden (2008) p.38 [2] Cohn (2003) p.425 [3] Fried & Jarden (2008) p.39 [4] Cohn (2003) p.426 [5] Fried & Jarden (2008) p.44 [6] Cohn (2003) p.427
• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.). Springer-Verlag. pp. 38–41. ISBN 978-3-540-77269-9. Zbl 1145.12001. • M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) 621
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• Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. ISBN 1-85233-587-4. Zbl 1003.00001. • A. Weil, Foundations of algebraic geometry.
Chapter 199
Regular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat’s Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers. The first few regular odd primes are: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ... (sequence A007703 in OEIS).
199.1 Definition 199.1.1
Class number criterion
An odd prime number p is defined to be regular if it does not divide the class number of the p-th cyclotomic field Q(ζp), where ζp is a p-th root of unity, it is listed on A000927. The prime number 2 is often considered regular as well. The class number of the cyclotomic field is the number of ideals of the ring of integers Z(ζp) up to isomorphism. Two ideals I,J are considered isomorphic if there is a nonzero u in Q(ζp) so that I=uJ.
199.1.2
Kummer’s criterion
Ernst Kummer (Kummer 1850) showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli numbers Bk for k = 2, 4, 6, …, p − 3. Kummer’s proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem, which states certain consequences of p dividing one of these Bernoulli numbers.
199.2 Siegel’s conjecture It has been conjectured that there are infinitely many regular primes. More precisely Carl Ludwig Siegel (1964) conjectured that e−1/2 , or about 60.65%, of all prime numbers are regular, in the asymptotic sense of natural density. Neither conjecture has been proven to date, 2015.
199.3 Irregular primes An odd prime that is not regular is an irregular prime. The first few irregular primes are: 623
624
CHAPTER 199. REGULAR PRIME 37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, ... (sequence A000928 in OEIS)
199.3.1
Infinitude
K. L. Jensen (an otherwise unknown student of Nielsen[1] ) proved in 1915 that there are infinitely many irregular primes of the form 4n + 3. [2] In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.[3] Metsänkylä proved[4] that for any integer T > 6, there are infinitely many irregular primes not of the form mT + 1 or mT − 1.
199.3.2
Irregular pairs
If p is an irregular prime and p divides the numerator of the Bernoulli number B₂k for 0 < 2k < p − 1, then (p, 2k) is called an irregular pair. In other words, an irregular pair is a book-keeping device to record, for an irregular prime p, the particular indices of the Bernoulli numbers at which regularity fails. The first few irregular pairs are: (691, 12), (3617, 16), (43867, 18), (283, 20), (617, 20), (131, 22), (593, 22), (103, 24), (2294797, 24), (657931, 26), (9349, 28), (362903, 28), ... (sequence A189683 in OEIS). The smallest even k such that nth irregular prime divides Bk are 32, 44, 58, 68, 24, 22, 130, 62, 84, 164, 100, 84, 20, 156, 88, 292, 280, 186, 100, 200, 382, 126, 240, 366, 196, 130, 94, 292, 400, 86, 270, 222, 52, 90, 22, ... (sequence A035112 in OEIS) For a given prime p, the number of such pairs is called the index of irregularity of p.[5] Hence, a prime is regular if and only if its index of irregularity is zero. Similarly, a prime is irregular if and only if its index of irregularity is positive. It was discovered that (p, p − 3) is in fact an irregular pair for p = 16843, as well as for p = 2124679. There are no more occurrences for p < 109 .
199.3.3
Irregular index
An odd prime p has irregular index n if and only if there are n values of k for which p divides B2k and these ks are less than (p − 1)/2. The first irregular prime with irregular index greater than 1 is 157, which divides B62 and B110 , so it has an irregular index 2. Clearly, the irregular index of a regular prime is 0. The irregular index of the nth prime is −1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, ... (This sequence defines “the irregular index of 2” as −1.) (sequence A091888 in OEIS) The irregular index of the nth irregular prime is 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, ... (sequence A091887 in OEIS) The primes having irregular index 1 are 37, 59, 67, 101, 103, 131, 149, 233, 257, 263, 271, 283, 293, 307, 311, 347, 389, 401, 409, 421, 433, 461, 463, 523, 541, 557, 577, 593, 607, 613, 619, 653, 659, 677, 683, 727, 751, 757, 761, 773, 797, 811, 821, 827, 839, 877, 881, 887, 953, 971, ... (sequence A073276 in OEIS)
199.3. IRREGULAR PRIMES
625
The primes having irregular index 2 are 157, 353, 379, 467, 547, 587, 631, 673, 691, 809, 929, 1291, 1297, 1307, 1663, 1669, 1733, 1789, 1933, 1997, 2003, 2087, 2273, 2309, 2371, 2383, 2423, 2441, 2591, 2671, 2789, 2909, 2957, ... (sequence A073277 in OEIS) The primes having irregular index 3 are 491, 617, 647, 1151, 1217, 1811, 1847, 2939, 3833, 4003, 4657, 4951, 6763, 7687, 8831, 9011, 10463, 10589, 12073, 13217, 14533, 14737, 14957, 15287, 15787, 15823, 16007, 17681, 17863, 18713, 18869, ... (sequence A060975 in OEIS) The least primes having irregular index n are 2, 3, 37, 157, 491, 12613, 78233, 527377, 3238481, ... (sequence A061576 in OEIS) (This sequence defines “the irregular index of 2” as −1, and also starts at n = −1.)
199.3.4
Euler irregular primes
Similarly, we can define an Euler irregular prime as a prime p that divides at least one E2n with 0 ≤ 2n ≤ p − 3. The first few Euler irregular primes are 19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587, ... (sequence A120337 in OEIS) The Euler irregular pairs are (61, 6), (277, 8), (19, 10), (2659, 10), (43, 12), (967, 12), (47, 14), (4241723, 14), (228135437, 16), (79, 18), (349, 18), (84224971, 18), (41737, 20), (354957173, 20), (31, 22), (1567103, 22), (1427513357, 22), (2137, 24), (111691689741601, 24), (67, 26), (61001082228255580483, 26), (71, 28), (30211, 28), (2717447, 28), (77980901, 28), ... Vandiver proved that Fermat’s Last Theorem (xp + yp = zp ) has no solution for integers x, y, z with gcd(xyz, p) = 1 if p is Euler-regular. Gut proved that x2p + y2p = z2p has no solution if p has an E-irregularity index less than 5. It was proven that there is an infinity of E-irregular primes. A stronger result was obtained: there is an infinity of E-irregular primes congruent to 1 modulo 8. As in the case of Kummer’s B-regular primes, there is as yet no proof that there are infinitely many E-regular primes, though this seems likely to be true.
199.3.5
Strong Irregular primes
A prime p is strong irregular if and only if p divides the numerator of B2n for some n < p−1 2 , and p also divides , (the two ns can be either the same or different), where Bn is the E2n for some n < p−1 Bernoulli number and En is 2 the Euler number, the first few strong irregular primes are 67, 101, 149, 263, 307, 311, 353, 379, 433, 461, 463, 491, 541, 577, 587, 619, 677, 691, 751, 761, 773, 811, 821, 877, 887, 929, 971, 1151, 1229, 1279, 1283, 1291, 1307, 1319, 1381, 1409, 1429, 1439, ... (sequence A128197 in OEIS) Clearly, to prove the Fermat’s last theorem for a strong irregular prime p is more difficult, the most difficult is that p is not only a strong irregular prime, but 2p+1, 4p+1, 8p+1, 10p+1, 14p+1, and 16p+1 are also all composite, the first case is that p = 263.
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CHAPTER 199. REGULAR PRIME
199.3.6
Weak irregular primes
A prime p is weak irregular if and only if p divides the numerator of B2n or E2n for some n < the Bernoulli number and En is the Euler number, the first few weak irregular primes are
p−1 2
, where Bn is
19, 31, 37, 43, 47, 59, 61, 67, 71, 79, 101, 103, 131, 137, 139, 149, 157, 193, 223, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 347, 349, 353, 373, 379, 389, 401, 409, 419, 421, 433, 461, 463, 491, 509, 523, 541, 547, 557, 563, 571, 577, 587, 593, ... The first values of Bernoulli and Euler numbers are 1, 1, 1, 1, 1, 5, 1, 61, 1, 1385, 5, 50521, 691, 2702765, 7, 199360981, 3617, 19391512145, 43867, 2404879675441, 174611, 370371188237525, 854513, 69348874393137901, 236364091, 15514534163557086905, 8553103, 4087072509293123892361, 23749461029, 1252259641403629865468285, ... (a2n = the absolute value of the numerator of B2n, and a2n+1 = the absolute value of E2n, this sequence starts at n = 0) The weak irregular pairs are (For the pairs (p, n) which p divides an, and n ≤ p−2) (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ... (The even ns means the numerator of B , and the odd ns means E -₁) The following table shows all weak irregular primes below 300 (Weak irregular index is defined as “Bernoulli irregular index + Euler irregular index”) The only primes below 1000 with weak irregular index 3 are 307, 311, 353, 379, 577, 587, 617, 619, 647, 691, 751, and 929. Besides, 491 is the only prime below 1000 with weak irregular index 4, and all other odd primes below 1000 with weak irregular index 0, 1, or 2.
199.3.7
Harmonic irregular primes
A prime p such that p divides H for some 1≤k≤p−2 is called Harmonic irregular primes (since p (In fact, p2 ) always divides H -₁), where H is the numerator of the Harmonic numbers , the first of them are 11, 29, 37, 43, 53, 61, 97, 109, 137, 173, 199, 227, 257, 269, 271, 313, 347, 353, 379, 397, 401, 409, 421, 433, 439, 509, 521, 577, 599, ... (sequence A092194 in OEIS) The density of them is about 0.367879... ~~very close to that of B-irregular or E-irregular primes. The numerator of the Harmonic numbers (also called Wolstenholme numbers) are 0, 1, 3, 11, 25, 137, 49, 363, 761, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 14274301, 275295799, 55835135, 18858053, 19093197, 444316699, 1347822955, 34052522467, 34395742267, 312536252003, 315404588903, 9227046511387, ... (this sequence starts at n = 0) (sequence A001008 in OEIS) The Harmonic irregular pairs are (11, 3), (137, 5), (11, 7), (761, 8), (7129, 9), (61, 10), (97, 11), (863, 11), (509, 12), (29, 13), (43, 13), (919, 13), (1049, 14), (1117, 14), (29, 15), (41233, 15), (8431, 16), (37, 17), (1138979, 17), (39541, 18), (37, 19), (7440427, 19), ... In fact, if and only if a prime p divides H , then p also divides H -₁- , so all odd prime p have an even Harmonic irregular index (0 is also an even number). The super irregular primes (odd primes which is B-irregular, E-irregular, and H-irregular) are
199.4. HISTORY
627
353, 379, 433, 577, 677, 761, 773, 821, 929, 971, ... The super regular primes (odd primes which is B-regular, E-regular, and H-regular) are 3, 5, 7, 13, 17, 23, 41, 73, 83, 89, 107, 113, 127, 151, 163, 167, 179, 181, 191, 197, 211, 229, 239, 281, 317, 331, 337, 367, 383, 431, 443, 449, 457, 479, 487, 503, 569, ...
199.4 History In 1850, Kummer proved that Fermat’s Last Theorem is true for a prime exponent p if p is regular. This raised attention in the irregular primes.[6] In 1852, Genocchi was able to prove that the first case of Fermat’s Last Theorem is true for an exponent p, if (p, p − 3) is not an irregular pair. Kummer improved this further in 1857 by showing that for the “first case” of Fermat’s Last Theorem (see Sophie Germain’s theorem) it is sufficient to establish that either (p, p − 3) or (p, p − 5) fails to be an irregular pair. Kummer found the irregular primes less than 165. In 1963, Lehmer reported results up to 10000 and Selfridge and B. W. Pollack announced in 1964 to have completed the table of irregular primes up to 25000. Although the two latter tables did not appear in print, Johnson found that (p, p − 3) is in fact an irregular pair for p = 16843 and that this is the first and only time this occurs for p < 30000.[7] It was found in 1993 that the next time this happens is for p = 2124679, see Wolstenholme prime.[8]
199.5 See also • Wolstenholme prime
199.6 References [1] Leo Corry: Number Crunching vs. Number Theory: Computers and FLT, from Kummer to SWAC (1850-1960), and beyond [2] Jensen, K. L. (1915). “Om talteoretiske Egenskaber ved de Bernoulliske Tal”. Nyt Tidsskr. Mat. B 26: 73–83. [3] Carlitz, L. (1954). “Note on irregular primes” (PDF). Proceedings of the American Mathematical Society (AMS) 5: 329– 331. doi:10.1090/S0002-9939-1954-0061124-6. ISSN 1088-6826. MR 61124. [4] Tauno Metsänkylä (1971). “Note on the distribution of irregular primes”. Ann. Acad. Sci. Fenn. Ser. A I 492. MR 0274403. [5] Narkiewicz, Władysław (1990), Elementary and analytic theory of algebraic numbers (2nd, substantially revised and extended ed.), Springer-Verlag; PWN-Polish Scientific Publishers, p. 475, ISBN 3-540-51250-0, Zbl 0717.11045 [6] Gardiner, A. (1988), “Four Problems on Prime Power Divisibility”, American Mathematical Monthly 95 (10): 926–931, doi:10.2307/2322386 [7] Johnson, W. (1975), “Irregular Primes and Cyclotomic Invariants” (PDF), Mathematics of Computation 29 (129): 113–120, doi:10.2307/2005468 Archived at WebCite [8] J. Buhler, R. Crandall, R. Ernvall and T. Metsänkylä, “Irregular primes and cyclotomic invariants to four million”, Math. Comp. 61 (1993), 151–153.
199.7 Further reading • Kummer, E. E. (1850), “Allgemeiner Beweis des Fermat’schen Satzes, dass die Gleichung xλ + yλ = zλ durch ganze Zahlen unlösbar ist, für alle diejenigen Potenz-Exponenten λ, welche ungerade Primzahlen sind und in den Zählern der ersten (λ−3)/2 Bernoulli’schen Zahlen als Factoren nicht vorkommen”, J. Reine Angew. Math. 40: 131–138
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• Carl Ludwig Siegel (1964). “Zu zwei Bemerkungen Kummers”. Nachr. Akad. d. Wiss. Goettingen, Math. Phys. K1. II: 51–62. • Iwasawa, K.; Sims, C. C. (1966), “Computation of invariants in the theory of cyclotomic fields”, Journal of the Mathematical Society of Japan 18 (1): 86–96, doi:10.2969/jmsj/01810086 Archived at WebCite • Wagstaff, Jr., S. S. (1978), “The Irregular Primes to 125000” (PDF), Mathematics of Computation 32 (142): 583–591, doi:10.2307/2006167 Archived at WebCite • Granville, A.; Monagan, M. B. (1988), “The First Case of Fermat’s Last Theorem is True for All Prime Exponents up to 714,591,416,091,389” (PDF), Transactions of the American Mathematical Society 306 (1): 329–359, doi:10.1090/S0002-9947-1988-0927694-5, MR 0002994788archived at WebCite • Gardiner, A. (1988), “Four Problems on Prime Power Divisibility”, American Mathematical Monthly 95 (10): 926–931, doi:10.2307/2322386 • Ernvall, R.; Metsänkylä, T. (1991), “Cyclotomic Invariants for Primes Between 125000 and 150000” (PDF), Mathematics of Computation 56 (194): 851–858, doi:10.2307/2008413 Archived at WebCite • Ernvall, R.; Metsänkylä, T. (1992), “Cyclotomic Invariants for Primes to One Million” (PDF), Mathematics of Computation 59 (199): 249–250, doi:10.2307/2152994 • Buhler, J. P.; Crandall, R. E.; Sompolski, R. W. (1992), “Irregular Primes to One Million” (PDF), Mathematics of Computation 59 (200): 717–722, doi:10.2307/2153086 Archived at WebCite • Boyd, D. W. (1994). “A p-adic Study of the Partial Sums of the Harmonic Series”. Experimental Mathematics 3 (4): 287–302. doi:10.1080/10586458.1994.10504298. Zbl 0838.11015. CiteSeerX: 10.1.1.56.7026. • Shokrollahi, M. A. (1996), “Computation of Irregular Primes up to Eight Million (Preliminary Report)", ICSI Technical Report, TR-96-002, CiteSeerX: 10.1.1.38.4040Archived at WebCite • Buhler, J.; Crandall, R.; Ernvall, R.; Metsänkylä, T.; Shokrollahi, M.A. (2001), “Irregular Primes and Cyclotomic Invariants to 12 Million”, Journal of Symbolic Computation 31 (1-2): 89–96, doi:10.1006/jsco.1999.1011 • Richard K. Guy (2004). “Section D2. The Fermat Problem”. Unsolved Problems in Number Theory (3rd ed.). Springer Verlag. ISBN 0-387-20860-7. • Villegas, F. R. (2007). Experimental Number Theory. New York: Oxford University Press. pp. 166–167. ISBN 978-0-19-852822-7.
199.8 External links • Weisstein, Eric W., “Irregular prime”, MathWorld. • Chris Caldwell, The Prime Glossary: regular prime at The Prime Pages. • Keith Conrad, Fermat’s last theorem for regular primes. • Bernoulli irregular prime • Euler irregular prime • Bernoulli and Euler irregular primes. • Factorization of Bernoulli and Euler numbers
Chapter 200
Resolvent (Galois theory) In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are • X 2 − ∆ where ∆ is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; its roots appear explicitly in the formulas for the roots of a cubic equation. • The cubic resolvent of a quartic equation, which is a resolvent for the dihedral group of 8 elements. • The Cayley resolvent is a resolvent for the maximal resoluble Galois group in degree five. It is a polynomial of degree 6. These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible. It is not known if there is an always separable resolvent for every group of permutations. For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble.
200.1 Definition Let n be a positive integer, which will be the degree of the equation that we will consider, and (X1 , . . . , Xn ) an ordered list of indeterminates. This defines the generic polynomial of degree n
F (X) = X n +
n n ∑ ∏ (−1)i Ei X n−i = (X − Xi ), i=1
i=1
where Ei is the ith elementary symmetric polynomial. The symmetric group Sn acts on the Xi by permuting them, and this induces an action on the polynomials in the Xi. The stabilizer of a given polynomial under this action is generally trivial, but some polynomials have a bigger stabilizer. For example, the stabilizer of an elementary symmetric polynomial is the whole group Sn. If the stabilizer is non-trivial, the polynomial is fixed by some non-trivial subgroup G; it is said an invariant of G. Conversely, given a subgroup G of Sn, an invariant of G is a resolvent invariant for G if it is not an invariant of any bigger subgroup of Sn.[1] Finding invariants for a given subgroup G of Sn is relatively easy; one can sum the orbit of a monomial under the action of Sn. However it may occur that the resulting polynomial is an invariant for a larger group. For example, let 629
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CHAPTER 200. RESOLVENT (GALOIS THEORY)
us consider the case of the subgroup G of S4 of order 4, consisting of (12)(34), (13)(24), (14)(23) and the identity (for the notation, see Permutation group). The monomial X1 X2 gives the invariant 2(X1 X2 + X3 X4 ) . It is not a resolvent invariant for G, as being invariant by (12). In fact, it is a resolvent invariant for the dihedral subgroup ⟨(12), (1324)⟩ , and is used to define the cubic resolvent of the quartic equation. If P is a resolvent invariant for a group G of index m, then its orbit under Sn has order m. Let P1 , . . . , Pm be the elements of this orbit. Then the polynomial
RG =
m ∏
(Y − Pi )
i=1
is invariant under Sn. Thus, when expanded, its coefficients are polynomials in the Xi that are invariant under the action of the symmetry group and thus may be expressed as polynomials in the elementary symmetric polynomials. In other words, RG is an irreducible polynomial in Y whose coefficients are polynomial in the coefficients of F. Having the resolvent invariant as a root, it is called a resolvent (sometimes resolvent equation). Let us consider now an irreducible polynomial
f (X) = X n +
n ∑
ai X n−i =
i=1
n ∏
(X − xi ),
i=1
with coefficients in a given field K (typically the field of rationals) and roots xi in an algebraically closed field extension. (f ) Substituting the Xi by the xi and the coefficients of F by those of f in what precedes, we get a polynomial RG (Y ) , also called resolvent or specialized resolvent in case of ambiguity). If the Galois group of f is contained in G, the (f ) specialization of the resolvent invariant is invariant by G and is thus a root of RG (Y ) that belongs to K (is rational (f ) on K). Conversely, if RG (Y ) has a rational root, which is not a multiple root, the Galois group of f is contained in G.
200.2 Terminology There are some variants in the terminology. • Depending on the authors or on the context, resolvent may refer to resolvent invariant instead of to resolvent equation. • A Galois resolvent is a resolvent such that the resolvent invariant is linear in the roots. • The Lagrange resolvent may refer to the linear polynomial
n−1 ∑
Xi ω i
i=0
where ω is a primitive nth root of unity. It is the resolvent invariant of a Galois resolvent for the identity group. • A relative resolvent is defined similarly as a resolvent, but considering only the action of the elements of a given subgroup H of Sn, having the property that, if a relative resolvent for a subgroup G of H has a rational simple root and the Galois group of f is contained in H, then the Galois group of f is contained in G. In this context, a usual resolvent is called an absolute resolvent.
200.3. RESOLVENT METHOD
631
200.3 Resolvent method The Galois group of a polynomial of degree n is Sn or a proper subgroup of that. If a polynomial is irreducible, then the corresponding Galois group is a transitive subgroup. Transitive subgroups of Sn form a directed graph: one group can be a subgroup of several groups. One resolvent can tell if the Galois group of a polynomial is a (not necessarily proper) subgroup of given group. The resolvent method is just a systematic way to check groups one by one until only one group is possible. This does not mean that every group must be checked: every resolvent can cancel out many possible groups. For example for degree five polynomials there is never need for a resolvent of D5 : resolvents for A5 and M20 give desired information. One way is to begin from maximal (transitive) subgroups until the right one is found and then continue with maximal subgroups of that.
200.4 References [1] http://www.alexhealy.net/papers/math250a.pdf
• Dickson, Leonard E. (1959). Algebraic Theories. New York: Dover Publications Inc. p. ix+276. ISBN 0-486-49573-6. • Girstmair, K. (1983). “On the computation of resolvents and Galois groups”. Manuscripta Mathematica 43 (2–3): 289–307. doi:10.1007/BF01165834.
Chapter 201
Rigid analytic space In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. They were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of p-adic analytic manifolds, rigid analytic spaces admit meaningful notions of analytic continuation and connectedness. However, this comes at the cost of some conceptual complexity.
201.1 Definitions The basic rigid analytic object is the n-dimensional unit polydisc, whose ring of functions is the Tate algebra Tn, made of power series in n variables whose coefficients approach zero in some complete nonarchimedean field k. The Tate algebra is the completion of the polynomial ring in n variables under the Gauss norm (taking the supremum of coefficients), and the polydisc plays a role analogous to that of affine n-space in algebraic geometry. Points on the polydisc are defined to be maximal ideals in the Tate algebra, and if k is algebraically closed, these correspond to points in kn whose coordinates have size at most one. An affinoid algebra is a k-Banach algebra that is isomorphic to a quotient of the Tate algebra by an ideal. An affinoid is then a subset of the unit polydisc on which the elements of this ideal vanish, i.e., it is the set of maximal ideals containing the ideal in question. The topology on affinoids is subtle, using notions of affinoid subdomains (which satisfy a universality property with respect to maps of affinoid algebras) and admissible open sets (which satisfy a finiteness condition for covers by affinoid subdomains). In fact, the admissible opens in an affinoid do not in general endow it with the structure of a topological space, but they do form a Grothendieck topology (called the G-topology), and this allows one to define good notions of sheaves and gluing of spaces. A rigid-analytic space over k is a pair (X, OX ) describing a locally ringed G-topologized space with a sheaf of kalgebras, such that there is a covering by open subspaces isomorphic to affinoids. This is analogous to the notion of manifolds being coverable by open subsets isomorphic to euclidean space, or schemes being coverable by affines. Schemes over k can be analytified functorially, much like varieties over the complex numbers can be viewed as complex analytic spaces, and there is an analogous formal GAGA theorem. The analytification functor respects finite limits.
201.2 Other formulations Main article: Berkovich space Around 1970, Raynaud provided an interpretation of certain rigid analytic spaces as formal models, i.e., as generic fibers of formal schemes over the valuation ring R of k. In particular, he showed that the category of quasi-compact quasi-separated rigid spaces over k is equivalent to the localization of the category of quasi-compact admissible formal schemes over R with respect to admissible formal blow-ups. Here, a formal scheme is admissible if it is coverable by formal spectra of topologically finitely presented R algebras whose local rings are R-flat. 632
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633
Formal models suffer from a problem of uniqueness, since blow-ups allow more than one formal scheme to describe the same rigid space. Huber worked out a theory of adic spaces to resolve this, by taking a limit over all blow-ups. These spaces are quasi-compact, quasi-separated, and functorial in the rigid space, but lack a lot of nice topological properties. Vladimir Berkovich reformulated much of the theory of rigid analytic spaces in the late 1980s, using a generalization of the notion of Gelfand spectrum for commutative unital C*-algebras. The Berkovich spectrum of a Banach kalgebra A is the set of multiplicative semi-norms on A that are bounded with respect to the given norm on k, and it has a topology induced by evaluating these semi-norms on elements of A. Since the topology is pulled back from the real line, Berkovich spectra have many nice properties, such as compactness, path-connectedness, and metrizability. Many ring-theoretic properties are reflected in the topology of spectra, e.g., if A is Dedekind, then its spectrum is contractible. However, even very basic spaces tend to be unwieldy – the projective line over Cp is a compactification of the inductive limit of affine Bruhat–Tits buildings for PGL2 (F), as F varies over finite extensions of Qp, when the buildings are given a suitably coarse topology.
201.3 References • Non-Archimedean analysis by S. Bosch, U. Güntzer, R. Remmert ISBN 3-540-12546-9 • Conrad, Brian Several approaches to non-archimedean geometry lecture notes from the Arizona Winter School • Rigid Analytic Geometry and Its Applications (Progress in Mathematics) by Jean Fresnel, Marius van der Put ISBN 0-8176-4206-4 • Houzel, Christian (1966) [1995], “Espaces analytiques rigides (d'après R. Kiehl)", Séminaire Bourbaki, Exp. No. 327 10, Paris: Société Mathématique de France, pp. 215–235, MR 1610409 http://www.numdam.org/ item?id=SB_1966-1968__10__215_0 Missing or empty |title= (help) • Tate, John (1971) [1962], “Rigid analytic spaces”, Inventiones Mathematicae 12: 257–289, doi:10.1007/BF01403307, ISSN 0020-9910, MR 0306196 • Éléments de Géométrie Rigide. Volume I. Construction et étude géométrique des espaces rigides (Progress in Mathematics 286) by Ahmed Abbes, ISBN 978-3-0348-0011-2 • M. Raynaud, Géométrie analytique rigide d’après Tate, Kiehl,. . . Table ronde d’analyse non archimidienne, Bull. Soc. Math. Fr. Mém. 39/40 (1974), 319-327.
201.4 External links • Hazewinkel, Michiel, ed. (2001), “Rigid_analytic_space”, Encyclopedia of Mathematics, Springer, ISBN 9781-55608-010-4
Chapter 202
Ring class field In mathematics, a ring class field is the abelian extension of an algebraic number K field associated by class field theory to the ring class group of some order of the ring of integers of K.[1]
202.1 References [1] Frey, Gerhard; Lange, Tanja (2006), “Varieties over special fields”, Handbook of elliptic and hyperelliptic curve cryptography, Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, pp. 87–113, MR 2162721. See in particular p. 99.
202.2 External links • Ring class fields
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Ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with rational integer coefficients, xn + cn₋₁xn−1 + … + c0 . This ring is often denoted by OK or OK . Since any rational integer number belongs to K and is its integral element, the ring Z is always a subring of OK. The ring Z is the simplest possible ring of integers.[1] Namely, Z = OQ where Q is the field of rational numbers.[2] And indeed, in algebraic number theory the elements of Z are often called the “rational integers” because of this. The ring of integers of an algebraic number field is the unique maximal order in the field.
203.1 Properties The ring of integers OK is a finitely-generated Z-module. Indeed it is a free Z-module, and thus has an integral basis, that is a basis b1 , … ,bn ∈ OK of the Q-vector space K such that each element x in OK can be uniquely represented as
x=
n ∑
ai bi ,
i=1
with ai ∈ Z.[3] The rank n of OK as a free Z-module is equal to the degree of K over Q. The rings of integers in number fields are Dedekind domains.[4]
203.2 Examples If p is a prime, ζ is a pth root of unity and K = Q(ζ) is the corresponding cyclotomic field, then an integral basis of OK = Z[ζ] is given by (1, ζ, ζ2 , … , ζp−2 ).[5] If d is a square-free integer and K = Q(√d) is the corresponding quadratic field, then OK is a ring of quadratic integers and its integral basis is given by (1, (1 + √d)/2) if d ≡ 1 (mod 4) and by (1, √d) if d ≡ 2, 3 (mod 4).[6]
203.3 Multiplicative structure In a ring of integers, every element has a factorisation into irreducible elements, but the ring need not have the property of unique factorisation: for example, in the ring of integers ℤ[√−5] the element 6 has two essentially different factorisations into irreducibles:[7][4]
6 = 2 · 3 = (1 +
√ √ −5)(1 − −5) . 635
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A ring of integers is always a Dedekind domain, and so has unique factorisation of ideals into prime ideals.[8] The units of a ring of integers OK is a finitely generated abelian group by Dirichlet’s unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units.[9]
203.4 Generalization One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤1; this is a ring because of the strong triangle inequality.[10] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter’s ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.[2] For example, the p-adic integers Zp are the ring of integers of the p-adic numbers Qp .
203.5 References • Cassels, J.W.S. (1986). Local fields. London Mathematical Society Student Texts 3. Cambridge: Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006. • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859 • Samuel, Pierre (1972). Algebraic number theory. Hermann/Kershaw.
203.6 Notes [1] The ring of integers, without specifying the field, refers to the ring Z of “ordinary” integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra. [2] Cassels (1986) p.192 [3] Cassels (1986) p.193 [4] Samuel (1972) p.49 [5] Samuel (1972) p.43 [6] Samuel (1972) p.35 [7] Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0. [8] Samuel (1972) p.50 [9] Samuel (1972) pp.59-62 [10] Cassels (1986) p.41
Chapter 204
Fontaine’s period rings In mathematics, Fontaine’s period rings are a collection of commutative rings first defined by Jean-Marc Fontaine that are used to classify p-adic Galois representations.
204.1 The ring B R The ring BdR is defined as follows. Let Cp denote the completion of Qp . Let
˜ + = lim OCp /(p) E ←− x7→xp ˜ + is a sequence (x1 , x2 , · · · ) of elements xi ∈ OCp /(p) such that xp ≡ xi (mod p) . There is So an element of E i+1 ˜ + → OCp /(p) given by f (x1 , x2 , . . . ) = x1 . There is also a multiplicative (but not a natural projection map f : E i ˜ + → OCp defined by t(x, x2 , . . . ) = limi→∞ x additive) map t : E ˜pi , where the x ˜i are arbitrary lifts of the xi to OCp . The composite of t with the projection OCp → OCp /(p) is just f . The general theory of Witt vectors yields ˜ + ) → OCp such that θ([x]) = t(x) for all x ∈ E ˜ + , where [x] denotes the a unique ring homomorphism θ : W (E + + ˜ = W (E ˜ + )[1/p] with respect to Teichmüller representative of x . The ring BdR is defined to be completion of B ( ) + + ˜ → Cp . The field BdR is just the field of fractions of B . the ideal ker θ : B dR
204.2 References 204.2.1
Secondary sources
• Berger, Laurent (2004), “An introduction to the theory of p-adic representations”, Geometric aspects of Dwork theory I, Berlin: Walter de Gruyter GmbH & Co. KG, arXiv:math/0210184, ISBN 978-3-11-017478-6, MR 2023292 • Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory, retrieved 2010-02-05 • Fontaine, Jean-Marc, ed. (1994), Périodes p-adiques, Astérisque 223, Paris: Société Mathématique de France, MR 1293969
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Chapter 205
Rupture field In abstract algebra, a rupture field of a polynomial P (X) over a given field K such that P (X) ∈ K[X] is a field extension of K generated by a root a of P (X) .[1] √ For instance, if K = Q and P (X) = X 3 − 2 then Q[ 3 2] is a rupture field for P (X) . The notion is interesting mainly if P (X) is irreducible over K . In that case, all rupture fields of P (X) over K are isomorphic, non canonically, to KP = K[X]/(P (X)) : if L = K[a] where a is a root of P (X) , then the ring homomorphism f defined by f (k) = k for all k ∈ K and f (X mod P ) = a is an isomorphism. Also, in this case the degree of the extension equals the degree of P . A rupture √ field of a polynomial does not necessarily contain all the roots of that polynomial: √ √in the above example the field Q[ 3 2] does not contain the other two (complex) roots of P (X) (namely ω 3 2 and ω 2 3 2 where ω is a primitive third root of unity). For a field containing all the roots of a polynomial, see the splitting field.
205.1 Examples A rupture field of X 2 + 1 over R is C . It is also a splitting field. The rupture field of X 2 + 1 over F3 is F9 since there is no element of F3 with square equal to −1 (and all quadratic extensions of F3 are isomorphic to F9 ).
205.2 See also • Splitting field
205.3 References [1] Escofier, Jean-Paul (2001). Galois Theory. Springer. p. 62. ISBN 0-387-98765-7.
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Chapter 206
S-unit In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-units.
206.1 Definition Let K be a number field with ring of integers R. Let S be a finite set of prime ideals of R. An element x of K is an S-unit if the principal fractional ideal (x) is a product of primes in S (to positive or negative powers). For the ring of rational integers Z one may take S to be a finite set of prime numbers and define an S-unit to be a rational number whose numerator and denominator are divisible only by the primes in S.
206.2 Properties The S-units form a multiplicative group containing the units of R. Dirichlet’s unit theorem holds for S-units: the group of S-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to r + s, where r is the rank of the unit group and s = |S|.
206.3 S-unit equation The S-unit equation is a Diophantine equation u+v=1 with u, v restricted to being S-units of K. The number of solutions of this equation is finite and the solutions are effectively determined using estimates for linear forms in logarithms as developed in transcendence theory. A variety of Diophantine equations are reducible in principle to some form of the S-unit equation: a notable example is Siegel’s theorem on integral points on elliptic curves, and more generally superelliptic curves of the form yn =f(x).
206.4 References • Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs 104. Providence, RI: American Mathematical Society. pp. 19–22. ISBN 0-8218-3387-1. Zbl 1033.11006. • Lang, Serge (1978). Elliptic curves: Diophantine analysis. Grundlehren der mathematischen Wissenschaften 231. Springer-Verlag. pp. 128–153. ISBN 3-540-08489-4. 639
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• Lang, Serge (1986). Algebraic number theory. Springer-Verlag. ISBN 0-387-94225-4. Chap. V. • Smart, Nigel (1998). The algorithmic resolution of Diophantine equations. London Mathematical Society Student Texts 41. Cambridge University Press. Chap. 9. ISBN 0-521-64156-X. • Neukirch, Jürgen (1986). Class field theory. Grundlehren der mathematischen Wissenschaften 280. SpringerVerlag. pp. 72–73. ISBN 3-540-15251-2.
206.5 Further reading • Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs 9. Cambridge University Press. ISBN 978-0-521-88268-2. • Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.
Chapter 207
Separable extension In the subfield of algebra named field theory, a separable extension is an algebraic field extension E ⊃ F such that for every α ∈ E , the minimal polynomial of α over F is a separable polynomial (i.e., has distinct roots; see below for the definition in this context).[1] Otherwise, the extension is called inseparable. There are other equivalent definitions of the notion of a separable algebraic extension, and these are outlined later in the article. The importance of separable extensions lies in the fundamental role they play in Galois theory in finite characteristic. More specifically, a finite degree field extension is Galois if and only if it is both normal and separable.[2] Since algebraic extensions of fields of characteristic zero, and of finite fields, are separable, separability is not an obstacle in most applications of Galois theory.[3][4] For instance, every algebraic (in particular, finite degree) extension of the field of rational numbers is necessarily separable. Despite the ubiquity of the class of separable extensions in mathematics, its extreme opposite, namely the class of purely inseparable extensions, also occurs quite naturally. An algebraic extension E ⊃ F is a purely inseparable extension if and only if for every α ∈ E \ F , the minimal polynomial of α over F is not a separable polynomial (i.e., does not have distinct roots).[5] For a field F to possess a non-trivial purely inseparable extension, it must necessarily be an infinite field of prime characteristic (i.e. specifically, imperfect), since any algebraic extension of a perfect field is necessarily separable.[3]
207.1 Informal discussion An arbitrary polynomial f with coefficients in some field F is said to have distinct roots if and only if it has deg(f) roots in some extension field E ⊇ F . For instance, the polynomial g(X)=X2 +1 with real coefficients has precisely deg(g)=2 roots in the complex plane; namely the imaginary unit i, and its additive inverse −i, and hence does have distinct roots. On the other hand, the polynomial h(X)=(X−2)2 with real coefficients does not have distinct roots; only 2 can be a root of this polynomial in the complex plane and hence it has only one, and not deg(h)=2 roots. To test if a polynomial has distinct roots, it is not necessary to consider explicitly any field extension nor to compute the roots: a polynomial has distinct roots if and only if the greatest common divisor of the polynomial and its derivative is a constant. For instance, the polynomial g(X)=X2 +1 in the above paragraph, has 2X as derivative, and, over a field of characteristic different of 2, we have g(X) - (1/2 X) 2X = 1, which proves, by Bézout’s identity, that the greatest common divisor is a constant. On the other hand, over a field where 2=0, the greatest common divisor is g, and we have g(X) = (X+1)2 has 1=−1 as double root. On the other hand, the polynomial h does not have distinct roots, whichever is the field of the coefficients, and indeed, h(X)=(X−2)2 , its derivative is 2 (X−2) and divides it, and hence does have a factor of the form (X − α)2 for α = 2 ). Although an arbitrary polynomial with rational or real coefficients may not have distinct roots, it is natural to ask at this stage whether or not there exists an irreducible polynomial with rational or real coefficients that does not have distinct roots. The polynomial h(X)=(X−2)2 does not have distinct roots but it is not irreducible as it has a non-trivial factor (X−2). In fact, it is true that there is no irreducible polynomial with rational or real coefficients that does not have distinct roots; in the language of field theory, every algebraic extension of Q or R is separable and hence both of these fields are perfect. 641
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207.2 Separable and inseparable polynomials A polynomial f in F[X] is a separable polynomial if and only if every irreducible factor of f in F[X] has distinct roots.[6] The separability of a polynomial depends on the field in which its coefficients are considered to lie; for instance, if g is an inseparable polynomial in F[X], and one considers a splitting field, E, for g over F, g is necessarily separable in E[X] since an arbitrary irreducible factor of g in E[X] is linear and hence has distinct roots.[1] Despite this, a separable polynomial h in F[X] must necessarily be separable over every extension field of F.[7] Let f in F[X] be an irreducible polynomial and f' its formal derivative. Then the following are equivalent conditions for f to be separable; that is, to have distinct roots: • If E ⊇ F and α ∈ E , then (X − α)2 does not divide f in E[X].[8] • There exists K ⊇ F such that f has deg(f) roots in K.[8] • f and f' do not have a common root in any extension field of F.[9] • f' is not the zero polynomial.[10] By the last condition above, if an irreducible polynomial does not have distinct roots, its derivative must be zero. Since the formal derivative of a positive degree polynomial can be zero only if the field has prime characteristic, for an irreducible polynomial to not have distinct roots its coefficients must lie in a field of prime characteristic. More generally, if an irreducible (non-zero) polynomial f in F[X] does not have distinct roots, not only must the characteristic of F be a (non-zero) prime number p, but also f(X)=g(Xp ) for some irreducible polynomial g in F[X].[11] n By repeated application of this property, it follows that in fact, f (X) = g(X p ) for a non-negative integer n and some separable irreducible polynomial g in F[X] (where F is assumed to have prime characteristic p).[12] By the property noted in the above paragraph, if f is an irreducible (non-zero) polynomial with coefficients in the field F of prime p, and does not have distinct roots, it is possible to write f(X)=g(Xp ). Furthermore, ∑ characteristic i if = ai X , and if the Frobenius endomorphism F is an automorphism, g may be written as g(X) = ∑g(X) ∑ p pi of ∑ bpi X i , and in particular, f (X) = g(X p ) = bi X = ( bi X i )p ; a contradiction of the irreducibility of f. Therefore, if F[X] possesses an inseparable irreducible (non-zero) polynomial, then the Frobenius endomorphism of F cannot be an automorphism (where F is assumed to have prime characteristic p).[13] If K is a finite field of prime characteristic p, and if X is an indeterminant, then the field of rational functions over K, K(X), is necessarily imperfect. Furthermore, the polynomial f(Y)=Y p −X is inseparable.[1] (To see this, note that there is some extension field E ⊇ K(X) in which f has a root α ; necessarily, αp = X in E. Therefore, working over E, f (Y ) = Y p − X = Y p − αp = (Y − α)p (the final equality in the sequence follows from freshman’s dream), and f does not have distinct roots.) More generally, if F is any field of (non-zero) prime characteristic for which the Frobenius endomorphism is not an automorphism, F possesses an inseparable algebraic extension.[14] A field F is perfect if and only if all of its algebraic extensions are separable (in fact, all algebraic extensions of F are separable if and only if all finite degree extensions of F are separable). By the argument outlined in the above paragraphs, it follows that F is perfect if and only if F has characteristic zero, or F has (non-zero) prime characteristic p and the Frobenius endomorphism of F is an automorphism.
207.3 Properties • If E ⊇ F is an algebraic field extension, and if α, β ∈ E are separable over F, then α+β and αβ are separable over F. In particular, the set of all elements in E separable over F forms a field.[15] • If E ⊇ L ⊇ F is such that E ⊇ L and L ⊇ F are separable extensions, then E ⊇ F is separable.[16] Conversely, if E ⊇ F is a separable algebraic extension, and if L is any intermediate field, then E ⊇ L and L ⊇ F are separable extensions.[17] • If E ⊇ F is a finite degree separable extension, then it has a primitive element; i.e., there exists α ∈ E with E = F [α] . This fact is also known as the primitive element theorem or Artin’s theorem on primitive elements.
207.4. SEPARABLE EXTENSIONS WITHIN ALGEBRAIC EXTENSIONS
643
207.4 Separable extensions within algebraic extensions Separable extensions occur quite naturally within arbitrary algebraic field extensions. More specifically, if E ⊇ F is an algebraic extension and if S = {α ∈ E|α is separable over F } , then S is the unique intermediate field that is separable over F and over which E is purely inseparable.[18] If E ⊇ F is a finite degree extension, the degree [S : F] is referred to as the separable part of the degree of the extension E ⊇ F (or the separable degree of E/F), and is often denoted by [E : F] ₑ or [E : F] .[19] The inseparable degree of E/F is the quotient of the degree by the separable degree. When the characteristic of F is p > 0, it is a power of p.[20] Since the extension E ⊇ F is separable if and only if S = E , it follows that for separable extensions, [E : F]=[E : F] ₑ , and conversely. If E ⊇ F is not separable (i.e., inseparable), then [E : F] ₑ is necessarily a non-trivial divisor of [E : F], and the quotient is necessarily a power of the characteristic of F.[19] On the other hand, an arbitrary algebraic extension E ⊇ F may not possess an intermediate extension K that is purely inseparable over F and over which E is separable (however, such an intermediate extension does exist when E ⊇ F is a finite degree normal extension (in this case, K can be the fixed field of the Galois group of E over F)). If such an intermediate extension does exist, and if [E : F] is finite, then if S is defined as in the previous paragraph, [E : F] ₑ =[S : F]=[E : K].[21] One known proof of this result depends on the primitive element theorem, but there does exist a proof of this result independent of the primitive element theorem (both proofs use the fact that if K ⊇ F is a purely inseparable extension, and if f in F[X] is a separable irreducible polynomial, then f remains irreducible in K[X][22] ). The equality above ([E : F] ₑ =[S : F]=[E : K]) may be used to prove that if E ⊇ U ⊇ F is such that [E : F] is finite, then [E : F] ₑ =[E : U] ₑ [U : F] ₑ .[23] If F is any field, the separable closure F sep of F is the field of all elements in an algebraic closure of F that are separable over F. This is the maximal Galois extension of F. By definition, F is perfect if and only if its separable and algebraic closures coincide (in particular, the notion of a separable closure is only interesting for imperfect fields).
207.5 The definition of separable non-algebraic extension fields Although many important applications of the theory of separable extensions stem from the context of algebraic field extensions, there are important instances in mathematics where it is profitable to study (not necessarily algebraic) separable field extensions. Let F /k be a field extension and let p be the characteristic exponent of k .[24] For any field extension L of k, we write FL = L ⊗k F (cf. Tensor product of fields.) Then F is said to be separable over k if the following equivalent conditions are met: • F p and k are linearly disjoint over k p • Fk1/p is reduced. • FL is reduced for all field extensions L of k. (In other words, F is separable over k if F is a separable k-algebra.) A separating transcendence basis for F/k is an algebraically independent subset T of F such that F/k(T) is a finite separable extension. An extension E/k is separable if and only if every finitely generated subextension F/k of E/k has a separating transcendence basis.[25] Suppose there is some field extension L of k such that FL is a domain. Then F is separable over k if and only if the field of fractions of FL is separable over L. An algebraic element of F is said to be separable over k if its minimal polynomial is separable. If F /k is an algebraic extension, then the following are equivalent. • F is separable over k. • F consists of elements that are separable over k. • Every subextension of F/k is separable. • Every finite subextension of F/k is separable.
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If F /k is finite extension, then the following are equivalent. • (i) F is separable over k. • (ii) F = k(a1 , ..., ar ) where a1 , ..., ar are separable over k. • (iii) In (ii), one can take r = 1. • (iv) If K is an algebraic closure of k, then there are precisely [F : k] embeddings F into K which fix k. • (v) If K is any normal extension of k such that F embeds into K in at least one way, then there are precisely [F : k] embeddings F into K which fix k. In the above, (iii) is known as the primitive element theorem. Fix the algebraic closure k , and denote by ks the set of all elements of k that are separable over k. ks is then separable algebraic over k and any separable algebraic subextension of k is contained in ks ; it is called the separable closure of k (inside k ). k is then purely inseparable over ks . Put in another way, k is perfect if and only if k = ks .
207.6 Differential criteria The separability can be studied with the aid of derivations and Kähler differentials. Let F be a finitely generated field extension of a field k . Then
dimF Derk (F, F ) ≥ tr. degk F where the equality holds if and only if F is separable over k. In particular, if F /k is an algebraic extension, then Derk (F, F ) = 0 if and only if F /k is separable.[26] Let D1 , ..., Dm be a basis of Derk (F, F ) and a1 , ..., am ∈ F . Then F is separable algebraic over k(a1 , ..., am ) if and only if the matrix Di (aj ) is invertible. In particular, when m = tr. degk F , {a1 , ..., am } above is called the separating transcendence basis.
207.7 See also • Purely inseparable extension • Perfect field • Primitive element theorem • Normal extension • Galois extension • Algebraic closure
207.8 Notes [1] Isaacs, p. 281 [2] Isaacs, Theorem 18.13, p. 282 [3] Isaacs, Theorem 18.11, p. 281 [4] Isaacs, p. 293 [5] Isaacs, p. 298
207.9. REFERENCES
645
[6] Isaacs, p. 280 [7] Isaacs, Lemma 18.10, p. 281 [8] Isaacs, Lemma 18.7, p. 280 [9] Isaacs, Theorem 19.4, p. 295 [10] Isaacs, Corollary 19.5, p. 296 [11] Isaacs, Corollary 19.6, p. 296 [12] Isaacs, Corollary 19.9, p. 298 [13] Isaacs, Theorem 19.7, p. 297 [14] Isaacs, p. 299 [15] Isaacs, Lemma 19.15, p. 300 [16] Isaacs, Corollary 19.17, p. 301 [17] Isaacs, Corollary 18.12, p. 281 [18] Isaacs, Theorem 19.14, p. 300 [19] Isaacs, p. 302 [20] Lang 2002, Corollary V.6.2 [21] Isaacs, Theorem 19.19, p. 302 [22] Isaacs, Lemma 19.20, p. 302 [23] Isaacs, Corollary 19.21, p. 303 [24] The characteristic exponent of k is 1 if k has characteristic zero; otherwise, it is the characteristic of k. [25] Fried & Jarden (2008) p.38 [26] Fried & Jarden (2008) p.49
207.9 References • Borel, A. Linear algebraic groups, 2nd ed. • P.M. Cohn (2003). Basic algebra • Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. • I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-53419002-2. • Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University of Chicago Press. pp. 55–59. ISBN 0-226-42451-0. Zbl 1001.16500. • M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) • Silverman, Joseph (1993). The Arithmetic of Elliptic Curves. Springer. ISBN 0-387-96203-4.
207.10 External links • Hazewinkel, Michiel, ed. (2001), “separable extension of a field k”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 208
Separable polynomial In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of its distinct roots is equal to its degree.[1] This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In general, P(X) is separable if and only if it is square-free over any field that contains K, which holds if and only if P(X) is coprime to its formal derivative P′(X).
208.1 Older definition In an older definition, P(X) was considered separable if each of its irreducible factors in K[X] is separable in the modern definition[2] In this definition, separability depended on the field K, for example, any polynomial over a perfect field would have been considered separable. This definition, although it can be convenient for Galois theory, is no longer in use.
208.2 Separable field extensions Separable polynomials are used to define separable extensions: A field extension K ⊂ L is a separable extension if and only if for every α ∈ L , which is algebraic over K, the minimal polynomial of α over K is a separable polynomial. Inseparable extensions (that is extensions which are not separable) may occur only in characteristic p. The criterion above leads to the quick conclusion that if P is irreducible and not separable, then P′(X)=0. Thus we must have P(X) = Q(Xp ) for some polynomial Q over K, where the prime number p is the characteristic. With this clue we can construct an example: P(X) = Xp − T with K the field of rational functions in the indeterminate T over the finite field with p elements. Here one can prove directly that P(X) is irreducible, and not separable. This is actually a typical example of why inseparability matters; in geometric terms P represents the mapping on the projective line over the finite field, taking co-ordinates to their pth power. Such mappings are fundamental to the algebraic geometry of finite fields. Put another way, there are coverings in that setting that cannot be 'seen' by Galois theory. (See radical morphism for a higher-level discussion.) If L is the field extension K(T 1/p ), 646
208.3. APPLICATIONS IN GALOIS THEORY
647
in other words the splitting field of P, then L/K is an example of a purely inseparable field extension. It is of degree p, but has no automorphism fixing K, other than the identity, because T 1/p is the unique root of P. This shows directly that Galois theory must here break down. A field such that there are no such extensions is called perfect. That finite fields are perfect follows a posteriori from their known structure. One can show that the tensor product of fields of L with itself over K for this example has nilpotent elements that are non-zero. This is another manifestation of inseparability: that is, the tensor product operation on fields need not produce a ring that is a product of fields (so, not a commutative semisimple ring). If P(x) is separable, and its roots form a group (a subgroup of the field K), then P(x) is an additive polynomial.
208.3 Applications in Galois theory Separable polynomials occur frequently in Galois theory. For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divide the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable (which is the case for every p but a finite number) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P. Another example: P being as above, a resolvent R for a group G is a polynomial whose coefficients are polynomials in the coefficients of P, which provides some information on the Galois group of P. More precisely, if R is separable and has a rational root then the Galois group of P is contained in G. For example, if D is the discriminant of P then X 2 − D is a resolvent for the alternating group. This resolvent is always separable (assuming the characteristic is not 2) if P is irreducible, but most resolvents are not always separable.
208.4 See also • Frobenius endomorphism
208.5 References [1] S. Lang, Algebra, p. 178 [2] N. Jacobson, Basic Algebra I, p. 233
• Pages 240-241 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001
Chapter 209
Serre’s conjecture II (algebra) Not to be confused with the Serre conjecture in number theory or the Quillen–Suslin theorem, which is sometimes also referred to as Serre’s conjecture. In mathematics, Jean-Pierre Serre conjectured[1][2] the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H 1 (F, G) is zero. The conjecture holds in the case where F is a local field (such as p-adic field) or a global field with no real embeddings (such as Q(√−1)). This is a special case of the Kneser–Harder–Chernousov Hasse Principle for algebraic groups over global fields. (Note that such fields do indeed have cohomological dimension at most 2.[2] ) The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2.[3] The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Suslin theorem.[4] Building on this result, the conjecture holds if G is a classical group.[5] The conjecture also holds if G is one of certain kinds of exceptional group.[6]
209.1 References [1] Serre, J-P. (1962). “Cohomologie galoisienne des groupes algébriques linéaires”. Colloque sur la théorie des groupes algébriques: 53–68. [2] Serre, J-P. (1964). Cohomologie galoisienne. Lecture Notes in Mathematics 5. Springer. [3] de Jong, A.J.; He, Xuhua; Starr, Jason Michael. “Families of rationally simply connected varieties over surfaces and torsors for semisimple groups”. arXiv:0809.5224. [4] Merkurjev, A.S.; Suslin, A.A. (1983). “K-cohomology of Severi-Brauer varieties and the norm-residue homomorphism”. Math. USSR Izvestiya 21: 307–340. doi:10.1070/im1983v021n02abeh001793. [5] Bayer-Fluckiger, E.; Parimala, R. (1995). “Galois cohomology of the classical groups over fields of cohomological dimension ≤ 2”. Inventiones Mathematicae 122: 195–229. doi:10.1007/BF01231443. [6] Gille, P. (2001). “Cohomologie galoisienne des groupes algebriques quasi-déployés sur des corps de dimension cohomologique ≤ 2”. Compositio Mathematica 125: 283–325.
209.2 External links • Philippe Gille’s survey of the conjecture
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Chapter 210
Simple extension In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite simple extensions.
210.1 Definition A field extension L/K is called a simple extension if there exists an element θ in L with
L = K(θ). The element θ is called a primitive element, or generating element, for the extension; we also say that L is generated over K by θ. Every finite field is a simple extension of the prime field of the same characteristic. More precisely, if p is a prime number and q = pd the field Fq of q elements is a simple extension of degree d of Fp . This means that it is generated by an element θ which is a root of an irreducible polynomial of degree d. However, in this case, θ is normally not referred to as a primitive element. In fact, a primitive element of a finite field is usually defined as a generator of the field’s multiplicative group. More precisely, by little Fermat theorem, the nonzero elements of Fq (i.e. its multiplicative group) are the roots of the equation
xq−1 − 1 = 0, that is the (q−1)-th roots of unity. Therefore, in this context, a primitive element is a primitive (q−1)-th root of unity, that is a generator of the multiplicative group of the nonzero elements of the field. Clearly, a group primitive element is a field primitive element, but the contrary is false. Thus the general definition requires that every element of the field may be expressed as a polynomial in the generator, while, in the realm of finite fields, every nonzero element of the field is a pure power of the primitive element. To distinguish these meanings one may use field primitive element of L over K for the general notion, and group primitive element for the finite field notion.[1]
210.2 Structure of simple extensions If L is a simple extension of K generated by θ, it is the only field contained in L which contains both K and θ. This means that every element of L can be obtained from the elements of K and θ by finitely many field operations (addition, subtraction, multiplication and division). 649
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CHAPTER 210. SIMPLE EXTENSION
Let us consider the polynomial ring K[X]. One of its main properties is that there exists a unique ring homomorphism φ : K[X] → L p(X) 7→ p(θ) . Two cases may occur. If φ is injective, it may be extended to the field of fractions K(X) of K[X]. As we have supposed that L is generated by θ, this implies that φ is an isomorphism from K(X) onto L. This implies that every element of L is equal to an irreducible fraction of polynomials in θ, and that two such irreducible fractions are equal if and only if one may pass from one to the other by multiplying the numerator and the denominator by the same non zero element of K. If φ is not injective, let p(X) be a generator of its kernel, which is thus the minimal polynomial of θ. The image of φ is a subring of L, and thus an integral domain. This implies that p is an irreducible polynomial, and thus that the quotient ring K[X]/⟨p⟩ is a field. As L is generated by θ, φ is surjective, and φ induces an isomorphism from K[X]/⟨p⟩ onto L. This implies that every element of L is equal to a unique polynomial in θ, of degree lower than the degree of the extension.
210.3 Examples • C:R (generated by i) • Q(√2):Q (generated by √2), more generally (i.e., a√ finite extension of Q) is a simple extension √ √ any number field √ Q(α) for some α. For example, Q( 3, 7) is generated by 3 + 7 . • F(X):F (generated by X).
210.4 References [1] (Roman 1995)
• Roman, Steven (1995). Field Theory. Graduate Texts in Mathematics 158. New York: Springer-Verlag. ISBN 0-387-94408-7. Zbl 0816.12001.
Chapter 211
Skolem–Mahler–Lech theorem In additive number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers is generated by a linear recurrence relation, then with finitely many exceptions the positions at which the sequence is zero form a regularly repeating pattern. More precisely, this set of positions can be decomposed into the union of a finite set and finitely many full arithmetic progressions. Here, a full arithmetic progression means a set of all non-negative integers of the form ax + b where a and b are a fixed pair of integers that define the progression and x ranges over all integers that make this expression non-negative. This result is named after Thoralf Skolem, Kurt Mahler, and Christer Lech. Its proofs use p-adic analysis.
211.1 Example Consider the sequence 0, 0, 1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, 0, ... that alternates between zeros and the Fibonacci numbers. This sequence can be generated by the linear recurrence relation F (i) = F (i − 2) + F (i − 4) (a modified form of the Fibonacci recurrence), starting from the base cases F(0) = F(1) = F(3) = 0 and F(2) = 1. For this sequence, F(i) = 0 if and only if i is either zero or odd. Thus, the positions at which the sequence is zero can be partitioned into a finite set (the singleton set {0}) and a full arithmetic progression (the non-negative odd numbers). In this example, only one arithmetic progression was needed, but other recurrence sequences may have zeros at positions forming multiple arithmetic progressions.
211.2 References • Lech, C. (1953), “A Note on Recurring Series”, Arkiv der Mathematik 2: 417–421, doi:10.1007/bf02590997. • Mahler, K. (1956), “On the Taylor coefficients of rational functions”, Proc. Cambridge Philos. Soc. 52: 39–48, doi:10.1017/s0305004100030966. • Mahler, K. (1957), “Addendum to the paper “On the Taylor coefficients of rational functions"", Proc. Cambridge Philos. Soc. 53: 544, doi:10.1017/s0305004100032552.
211.3 External links • Weisstein, Eric W., “Skolem-Mahler-Lech Theorem”, MathWorld.
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Chapter 212
Splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.
212.1 Definition A splitting field of a polynomial p(X) over a field K is a field extension L of K over which p factors into linear factors p(X) =
∏deg(p) i=1
(X − ai ) where for each i we have (X − ai ) ∈ L[X]
and such that the roots ai generate L over K. The extension L is then an extension of minimal degree over K in which p splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known as the Galois group of p (if we assume it is separable).
212.2 Facts An extension L which is a splitting field for a set of polynomials p(X) over K is called a normal extension of K. Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p. If K is a subfield of the complex numbers, the existence is immediate. On the other hand, the existence of algebraic closures in general is often proved by 'passing to the limit' from the splitting field result, which therefore requires an independent proof to avoid circular reasoning. Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′ that is minimal, in an obvious sense. Such a Galois closure should contain a splitting field for all the polynomials p over K that are minimal polynomials over K of elements a of K′.
212.3 Constructing splitting fields 212.3.1
Motivation
Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, have no roots such as X2 +1 over R, the real numbers. By constructing the splitting field for such a polynomial one can find the roots of the polynomial in the new field.
212.3.2
The construction
Let F be a field and p(X) be a polynomial in the polynomial ring F[X] of degree n. The general process for constructing K, the splitting field of p(X) over F, is to construct a sequence of fields F = K0 , K1 , . . . Kr−1 , Kr = K such that 652
212.3. CONSTRUCTING SPLITTING FIELDS
653
Ki is an extension of Ki₋₁ containing a new root of p(X). Since p(X) has at most n roots the construction will require at most n extensions. The steps for constructing Ki are given as follows: • Factorize p(X) over Ki into irreducible factors f1 (X)f2 (X) · · · fk (X) . • Choose any nonlinear irreducible factor f(X) = fi(X). • Construct the field extension Ki₊₁ of Ki as the quotient ring Ki₊₁ = Ki[X]/(f(X)) where (f(X)) denotes the ideal in Ki[X] generated by f(X) • Repeat the process for Ki₊₁ until p(X) completely factors. The irreducible factor fi used in the quotient construction may be chosen arbitrarily. Although different choices of factors may lead to different subfield sequences the resulting splitting fields will be isomorphic. Since f(X) is irreducible, (f(X)) is a maximal ideal and hence Ki[X]/(f(X)) is, in fact, a field. Moreover, if we let π : Ki [X] → Ki [X]/(f (X)) be the natural projection of the ring onto its quotient then f (π(X)) = π(f (X)) = f (X) mod f (X) = 0 so π(X) is a root of f(X) and of p(X). The degree of a single extension [Ki+1 : Ki ] is equal to the degree of the irreducible factor f(X). The degree of the extension [K : F] is given by [Kr : Kr−1 ] · · · [K2 : K1 ][K1 : F ] and is at most n!.
212.3.3
The field Ki[X]/(f(X))
As mentioned above, the quotient ring Ki₊₁ = Ki[X]/(f(X)) is a field when f(X) is irreducible. Its elements are of the form cn−1 αn−1 + cn−2 αn−2 + · · · + c1 α + c0 where the cj are in Ki and α = π(X). (If one considers Ki₊₁ as a vector space over Ki then the powers αj for 0 ≤ j ≤ n−1 form a basis.) The elements of Ki₊₁ can be considered as polynomials in α of degree less than n. Addition in Ki₊₁ is given by the rules for polynomial addition and multiplication is given by polynomial multiplication modulo f(X). That is, for g(α) and h(α) in Ki₊₁ the product g(α)h(α) = r(α) where r(X) is the remainder of g(X)h(X) divided by f(X) in Ki[X]. The remainder r(X) can be computed through long division of polynomials, however there is also a straightforward reduction rule that can be used to compute r(α) = g(α)h(α) directly. First let f (X) = X n + bn−1 X n−1 + · · · + b1 X + b0 . The polynomial is over a field so one can take f(X) to be monic without loss of generality. Now α is a root of f(X), so αn = −(bn−1 αn−1 + · · · + b1 α + b0 ). If the product g(α)h(α) has a term αm with m ≥ n it can be reduced as follows: ( ) ( ) αn αm−n = − bn−1 αn−1 + · · · + b1 α + b0 αm−n = − bn−1 αm−1 + · · · + b1 αm−n+1 + b0 αm−n As an example of the reduction rule, take Ki = Q[X], the ring of polynomials with rational coefficients, and take f(X) = X7 − 2. Let g(α) = α5 + α2 and h(α) = α3 +1 be two elements of Q[X]/(X7 − 2). The reduction rule given by f(X) is α7 = 2 so ( )( ) ( ) g(α)h(α) = α5 + α2 α3 + 1 = α8 + 2α5 + α2 = α7 α + 2α5 + α2 = 2α5 + α2 + 2α.
654
CHAPTER 212. SPLITTING FIELD
212.4 Examples 212.4.1
The complex numbers
Consider the polynomial ring R[x], and the irreducible polynomial x2 + 1. The quotient ring R[x] / (x2 + 1) is given by the congruence x2 ≡ −1. As a result, the elements (or equivalence classes) of R[x] / (x2 + 1) are of the form a + bx where a and b belong to R. To see this, note that since x2 ≡ −1 it follows that x3 ≡ −x, x4 ≡ 1, x5 ≡ x, etc.; and so, for example p + qx + rx2 + sx3 ≡ p + qx + r⋅(−1) + s⋅(−x) = (p − r) + (q − s)⋅x. The addition and multiplication operations are given by firstly using ordinary polynomial addition and multiplication, but then reducing modulo x2 + 1, i.e. using the fact that x2 ≡ −1, x3 ≡ −x, x4 ≡ 1, x5 ≡ x, etc. Thus:
(a1 + b1 x) + (a2 + b2 x) = (a1 + a2 ) + (b1 + b2 )x, (a1 + b1 x)(a2 + b2 x) = a1 a2 + (a1 b2 + b1 a2 )x + (b1 b2 )x2 ≡ (a1 a2 − b1 b2 ) + (a1 b2 + b1 a2 )x . If we identify a + bx with (a,b) then we see that addition and multiplication are given by
(a1 , b1 ) + (a2 , b2 ) = (a1 + a2 , b1 + b2 ), (a1 , b1 ) · (a2 , b2 ) = (a1 a2 − b1 b2 , a1 b2 + b1 a2 ). We claim that, as a field, the quotient R[x] / (x2 + 1) is isomorphic to the complex numbers, C. A general complex number is of the form a + ib, where a and b are real numbers and i2 = −1. Addition and multiplication are given by
(a1 + ib1 ) + (a2 + ib2 ) = (a1 + a2 ) + i(b1 + b2 ), (a1 + ib1 ) · (a2 + ib2 ) = (a1 a2 − b1 b2 ) + i(a1 b2 + a2 b1 ). If we identify a + ib with (a,b) then we see that addition and multiplication are given by
(a1 , b1 ) + (a2 , b2 ) = (a1 + a2 , b1 + b2 ), (a1 , b1 ) · (a2 , b2 ) = (a1 a2 − b1 b2 , a1 b2 + b1 a2 ) . The previous calculations show that addition and multiplication behave the same way in R[x] / (x2 + 1) and C. In fact, we see that the map between R[x]/(x2 + 1) and C given by a + bx → a + ib is a homomorphism with respect to addition and multiplication. It is also obvious that the map a + bx → a + ib is both injective and surjective; meaning that a + bx → a + ib is a bijective homomorphism, i.e. an isomorphism. It follows that, as claimed: R[x] / (x2 + 1) ≅ C.
212.4.2
Cubic example
Let K be the rational number field Q and p(X) = X3 − 2. Each root of p equals
ω1 = 1 √ 3 1 i, ω2 = − + 2 2 √ 1 3 ω3 = − − i. 2 2
√ 3 2 times a cube root of unity. Therefore, if we denote the cube roots of unity by
212.5. SEE ALSO
655
any field containing two distinct roots of p will contain the quotient between two distinct cube roots of unity. Such a quotient is a primitive cube root of unity—either ω2 or ω3 = 1/ω2 . It follows that a splitting field L of p will contain ω2 , as well as the real cube root of 2; conversely, any extension of Q containing these elements contains all the roots of p. Thus √ √ √ √ √ 3 3 3 3 3 L = Q( 2, ω2 ) = {a + bω2 + c 2 + d 2ω2 + e 22 + f 22 ω2 | a, b, c, d, e, f ∈ Q}
212.4.3
Other examples
• The splitting field of x2 + 1 over F7 is F49 ; the polynomial has no roots in F7 , i.e., −1 is not a square there, because 7 is not equivalent to 1 (mod 4).[1] • The splitting field of x2 − 1 over F7 is F7 since x2 − 1 = (x + 1)(x − 1) already factors into linear factors. • We calculate the splitting field of f(x) = x3 + x + 1 over F2 . It is easy to verify that f(x) has no roots in F2 , hence f(x) is irreducible in F2 [x]. Put r = x + (f(x)) in F2 [x]/(f(x)) so F2 (r) is a field and x3 + x + 1 = (x + r)(x2 + ax + b) in F2 (r)[x]. Note that we can write + for − since the characteristic is two. Comparison of coefficients shows that a = r and b = 1 + r2 . The elements of F2 (r) can be listed as c + dr + er2 , where c, d, e are in F2 . There are eight elements: 0, 1, r, 1 + r, r2 , 1 + r2 , r + r2 and 1 + r + r2 . Substituting these in x2 + rx + 1 + r2 we reach (r2 )2 + r(r2 ) + 1 + r2 = r4 + r3 + 1 + r2 = 0, therefore x3 + x + 1 = (x + r)(x + r2 )(x + (r + r2 )) for r in F2 [x]/(f(x)); E = F2 (r) is a splitting field of x3 + x + 1 over F2 .
212.5 See also • Rupture field
212.6 Notes [1] Instead of applying this characterization of odd prime moduli for which −1 is a square, one could just check that the set of squares in F7 is the set of classes of 0, 1, 4, and 2, which does not include the class of −1≡6.
212.7 References • Dummit, David S., and Foote, Richard M. (1999). Abstract Algebra (2nd ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-36857-1. • Hazewinkel, Michiel, ed. (2001), “Splitting field of a polynomial”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Weisstein, Eric W., “Splitting field”, MathWorld.
Chapter 213
Splitting of prime ideals in Galois extensions In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of G need be considered, rather than two. This was certainly familiar before Hilbert.
213.1 Definitions Let L / K be a finite extension of number fields, and let B and A be the corresponding ring of integers of L and K, respectively, which are defined to be the integral closure of the integers Z in the field in question. A ↓ K
,→ ,→
B ↓ L
Finally, let p be a non-zero prime ideal in A, or equivalently, a maximal ideal, so that the residue A/p is a field. From the basic theory of one-dimensional rings follows the existence of a unique decomposition
pB =
∏
e(j)
Pj
j
of the ideal pB generated in B by p into a product of distinct maximal ideals Pj, with multiplicities e(j). The multiplicities e(j) are called ramification indices of the extension at p. If they are all equal to 1, the field extension L/K is called unramified at p. If this is the case, by the Chinese remainder theorem, the quotient B/pB is a product of fields Fj = B/Pj.
213.1.1
The Galois situation
In the following, the extension L / K is assumed to be a Galois extension. Then the Galois group G acts transitively on the Pj. That is, the prime ideal factors of p in L form a single orbit under the automorphisms of L over K. From 656
213.2. EXAMPLE — THE GAUSSIAN INTEGERS
657
this and the unique factorisation theorem, it follows that e(j) = e is independent of j; something that certainly need not be the case for extensions that are not Galois. The basic relation then reads
pB = (
∏
213.1.2
Pj )e .
Facts
• Given an extension as above, it is unramified in all but finitely many points. • In the unramified case, because of the transitivity of the Galois group action, the fields Fj introduced above are all isomorphic, say, to the finite field F ′ containing
F = A/p A counting argument shows that [L : K]/[F ′ : F ] equals the number of prime factors of p in B. By the orbit-stabilizer formula this number is also equal to |G|/|D| where by definition D, the decomposition group of p, is the subgroup of elements of G sending a given Pj to itself. That is, since the degree of L/K and the order of G are equal by basic Galois theory, the order of the decomposition group D is the degree of the residue field extension F ′ /F. The theory of the Frobenius element goes further, to identify an element of D, for j given, which generates the Galois group of the finite field extension. • In the ramified case, there is the further phenomenon of inertia: the index e is interpreted as the extent to which elements of G are not seen in the Galois groups of any of the residue field extensions. Each decomposition group D, for a given Pj, contains an inertia group I consisting of the g in G that send Pj to itself, but induce the identity automorphism on
Fj = B/Pj In the geometric analogue, for complex manifolds or algebraic geometry over an algebraically closed field, the concepts of decomposition group and inertia group coincide. There, given a Galois ramified cover, all but finitely many points have the same number of preimages. The splitting of primes in extensions that are not Galois may be studied by using a splitting field initially, i.e. a Galois extension that is somewhat larger. For example cubic fields usually are 'regulated' by a degree 6 field containing them.
213.2 Example — the Gaussian integers This section describes the splitting of prime ideals in the field extension Q(i)/Q. That is, we take K = Q and L = Q(i), so OK is simply Z, and OL = Z[i] is the ring of Gaussian integers. Although this case is far from representative — after all, Z[i] has unique factorisation, and there aren't many quadratic fields with unique factorization — it exhibits many of the features of the theory. Writing G for the Galois group of Q(i)/Q, and σ for the complex conjugation automorphism in G, there are three cases to consider.
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213.2.1
CHAPTER 213. SPLITTING OF PRIME IDEALS IN GALOIS EXTENSIONS
The prime p = 2
The prime 2 of Z ramifies in Z[i]:
(2) = (1 + i)2 , so the ramification index here is e = 2. The residue field is
OL /(1 + i)OL which is the finite field with two elements. The decomposition group must be equal to all of G, since there is only one prime of Z[i] above 2. The inertia group is also all of G, since
a + bi ≡ a − bi modulo (1 + i) , for any integers a and b. In fact, 2 is the only prime that ramifies in Z[i], since every prime that ramifies must divide the discriminant of Z[i], which is −4.
213.2.2
Primes p ≡ 1 mod 4
Any prime p ≡ 1 mod 4 splits into two distinct prime ideals in Z[i]; this is a manifestation of Fermat’s theorem on sums of two squares. For example,
(13) = (2 + 3i)(2 − 3i) . The decomposition groups in this case are both the trivial group {1}; indeed the automorphism σ switches the two primes (2 + 3i) and (2 − 3i), so it cannot be in the decomposition group of either prime. The inertia group, being a subgroup of the decomposition group, is also the trivial group. There are two residue fields, one for each prime,
OL /(2 ± 3i)OL , which are both isomorphic to the finite field with 13 elements. The Frobenius element is the trivial automorphism; this means that
(a + bi)13 ≡ a + bi modulo (2 ± 3i) for any integers a and b.
213.2.3
Primes p ≡ 3 mod 4
Any prime p ≡ 3 mod 4 remains inert in Z[i]; that is, it does not split. For example, (7) remains prime in Z[i]. In this situation, the decomposition group is all of G, again because there is only one prime factor. However, this situation differs from the p = 2 case, because now σ does not act trivially on the residue field
OL /(7)OL , which is the finite field with 72 = 49 elements. For example, the difference between 1 + i and σ(1 + i) = 1 − i is 2i, which is certainly not divisible by 7. Therefore the inertia group is the trivial group {1}. The Galois group of
213.3. COMPUTING THE FACTORISATION
659
this residue field over the subfield Z/7Z has order 2, and is generated by the image of the Frobenius element. The Frobenius is none other than σ; this means that
(a + bi)7 ≡ a − bi modulo 7 , for any integers a and b.
213.2.4
Summary
213.3 Computing the factorisation Suppose that we wish to determine the factorisation of a prime ideal P of OK into primes of OL. We will assume that the extension L/K is a finite separable extension; the extra hypothesis of normality in the definition of Galois extension is not necessary. The following procedure (Neukirch, p47) solves this problem in many cases. The strategy is to select an integer θ in OL so that L is generated over K by θ (such a θ is guaranteed to exist by the primitive element theorem), and then to examine the minimal polynomial H(X) of θ over K; it is a monic polynomial with coefficients in OK. Reducing the coefficients of H(X) modulo P, we obtain a monic polynomial h(X) with coefficients in F, the (finite) residue field OK/P. Suppose that h(X) factorises in the polynomial ring F[X] as
h(X) = h1 (X)e1 · · · hn (X)en , where the hj are distinct monic irreducible polynomials in F[X]. Then, as long as P is not one of finitely many exceptional primes (the precise condition is described below), the factorisation of P has the following form:
P OL = Qe11 · · · Qenn , where the Qj are distinct prime ideals of OL. Furthermore, the inertia degree of each Qj is equal to the degree of the corresponding polynomial hj, and there is an explicit formula for the Qj:
Qj = P OL + hj (θ)OL , where hj denotes here a lifting of the polynomial hj to K[X]. In the Galois case, the inertia degrees are all equal, and the ramification indices e1 = ... = en are all equal. The exceptional primes, for which the above result does not necessarily hold, are the ones not relatively prime to the conductor of the ring OK[θ]. The conductor is defined to be the ideal
{y ∈ OL : yOL ⊆ OK [θ]}; it measures how far the order OK[θ] is from being the whole ring of integers (maximal order) OL. A significant caveat is that there exist examples of L/K and P such that there is no available θ that satisfies the above hypotheses (see for example [1] ). Therefore the algorithm given above cannot be used to factor such P, and more sophisticated approaches must be used, such as that described in.[2]
213.3.1
An example
Consider again the case of the Gaussian integers. We take θ to be the imaginary unit i, with minimal polynomial H(X) = X2 + 1. Since Z[ i ] is the whole ring of integers of Q( i ), the conductor is the unit ideal, so there are no exceptional primes.
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CHAPTER 213. SPLITTING OF PRIME IDEALS IN GALOIS EXTENSIONS
For P = (2), we need to work in the field Z/(2)Z, which amounts to factorising the polynomial X2 + 1 modulo 2:
X 2 + 1 = (X + 1)2
(mod 2).
Therefore there is only one prime factor, with inertia degree 1 and ramification index 2, and it is given by
Q = (2)Z[i] + (i + 1)Z[i] = (1 + i)Z[i]. The next case is for P = (p) for a prime p ≡ 3 mod 4. For concreteness we will take P = (7). The polynomial X2 + 1 is irreducible modulo 7. Therefore there is only one prime factor, with inertia degree 2 and ramification index 1, and it is given by
Q = (7)Z[i] + (i2 + 1)Z[i] = 7Z[i]. The last case is P = (p) for a prime p ≡ 1 mod 4; we will again take P = (13). This time we have the factorisation
X 2 + 1 = (X + 5)(X − 5)
(mod 13).
Therefore there are two prime factors, both with inertia degree and ramification index 1. They are given by
Q1 = (13)Z[i] + (i + 5)Z[i] = · · · = (2 + 3i)Z[i] and
Q2 = (13)Z[i] + (i − 5)Z[i] = · · · = (2 − 3i)Z[i].
213.4 External links • Splitting and ramification in number fields and Galois extensions at PlanetMath.org. • William Stein, A brief introduction to classical and adelic algebraic number theory
213.5 References [1] http://modular.math.washington.edu/papers/undergrad/decomp/decomp/node4.html [2] http://modular.math.washington.edu/papers/undergrad/decomp/decomp/node3.html
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
Chapter 214
Square class In abstract algebra, a square class of a field (mathematics) F is an element of the square class group, the quotient group F × /F ×2 of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square class is a subset of the nonzero elements (a coset of the multiplicative group) consisting of the elements of the form xy2 where x is some particular fixed element and y ranges over all nonzero field elements.[1] For instance, if F = R , the field of real numbers, then F × is just the group of all nonzero real numbers (with the multiplication operation) and F ×2 is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers.[1] Square classes are frequently studied in relation to the theory of quadratic forms.[2] The reason is that if V is an F -vector space and q : V → F is a quadratic form and v is an element of V such that q(v) = a ∈ F × , then for all u ∈ F × , q(uv) = au2 and thus it is sometimes more convenient to talk about the square classes which the quadratic form represents. Every element of the square class group is an involution. It follows that, if the number of square classes of a field is finite, it must be a power of two.[2]
214.1 References [1] Salzmann, H. (2007), The Classical Fields: Structural Features of the Real and Rational Numbers, Encyclopedia of Mathematics and its Applications 112, Cambridge University Press, p. 295, ISBN 9780521865166. [2] Szymiczek, Kazimierz (1997), Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms, Algebra, logic, and applications 7, CRC Press, pp. 29, 109, ISBN 9789056990763.
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Chapter 215
Stark conjectures In number theory, the Stark conjectures, introduced by Stark (1971, 1975, 1976, 1980) and later expanded by Tate (1984), give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The conjectures generalize the analytic class number formula expressing the leading coefficient of the Taylor series for the Dedekind zeta function of a number field as the product of a regulator related to S-units of the field and a rational number. When K/k is an abelian extension and the order of vanishing of the L-function at s = 0 is one, Stark gave a refinement of his conjecture, predicting the existence of certain S-units, called Stark units. Rubin (1996) and Cristian Dumitru Popescu gave extensions of this refined conjecture to higher orders of vanishing.
215.1 Formulation The Stark conjectures, in the most general form, predict that the leading coefficient of an Artin L-function is the product of a type of regulator, the Stark regulator, with an algebraic number. When the extension is abelian and the order of vanishing of an L-function at s = 0 is one, Stark’s refined conjecture predicts the existence of the Stark units, whose roots generate Kummer extensions of K that are abelian over the base field k (and not just abelian over K, as Kummer theory implies). As such, this refinement of his conjecture has theoretical implications for solving Hilbert’s twelfth problem. Also, it is possible to compute Stark units in specific examples, allowing verification of the veracity of his refined conjecture as well as providing an important computational tool for generating abelian extensions of number fields. In fact, some standard algorithms for computing abelian extensions of number fields involve producing Stark units that generate the extensions (see below)
215.2 Computation The first order zero conjectures are used in recent versions of the PARI/GP computer algebra system to compute Hilbert class fields of totally real number fields, and the conjectures provide one solution to Hilbert’s twelfth problem, which challenged mathematicians to show how class fields may be constructed over any number field by the methods of complex analysis.
215.3 Progress Stark’s principal conjecture has been proven in various special cases, including the case where the character defining the L-function takes on only rational values. Except when the base field is the field of rational numbers or an imaginary quadratic field, the abelian Stark conjectures are still unproved in number fields, and more progress has been made in function fields of an algebraic variety. Manin (2004) related Stark’s conjectures to noncommutative geometry of Alain Connes.[1] This provides a conceptual framework for studying the conjectures, although at the moment it is unclear whether Manin’s techniques will yield the actual proof. 662
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663
215.4 Notes [1] Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences 49 (Second ed.). p. 171. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.
215.5 References • Burns, David; Sands, Jonathan; Solomon, David, eds. (2004), Stark’s conjectures: recent work and new directions, Contemporary Mathematics 358, Providence, RI: American Mathematical Society, doi:10.1090/conm/358, ISBN 978-0-8218-3480-0, MR 2090725 • Manin, Yuri Ivanovich (2004), “Real multiplication and noncommutative geometry (ein Alterstraum)", in Piene, Ragni; Laudal, Olav Arnfinn, The legacy of Niels Henrik Abel, Berlin, New York: Springer-Verlag, pp. 685–727, ISBN 978-3-540-43826-7, MR 2077591 • Popescu, Cristian D. (1999), “On a refined Stark conjecture for function fields”, Compositio Mathematica 116 (3): 321–367, doi:10.1023/A:1000833610462, ISSN 0010-437X, MR 1691163 • Rubin, Karl (1996), “A Stark conjecture over Z for abelian L-functions with multiple zeros”, Université de Grenoble. Annales de l'Institut Fourier 46 (1): 33–62, ISSN 0373-0956, MR 1385509 • Stark, Harold M. (1971), “Values of L-functions at s = 1. I. L-functions for quadratic forms.”, Advances in Mathematics 7: 301–343, doi:10.1016/S0001-8708(71)80009-9, ISSN 0001-8708, MR 0289429 • Stark, Harold M. (1975), “L-functions at s = 1. II. Artin L-functions with rational characters”, Advances in Mathematics 17 (1): 60–92, doi:10.1016/0001-8708(75)90087-0, ISSN 0001-8708, MR 0382194 • Stark, H. M. (1977), “Class fields and modular forms of weight one”, in Serre, Jean-Pierre; Zagier, D. B., Modular Functions of One Variable V: Proceedings International Conference, University of Bonn, Sonderforschungsbereich Theoretische Mathematik, July 1976, Lecture Notes in Math 601, Berlin, New York: Springer-Verlag, pp. 277–287, doi:10.1007/BFb0063951, ISBN 978-3-540-08348-1, MR 0450243 • Stark, Harold M. (1976), “L-functions at s = 1. III. Totally real fields and Hilbert’s twelfth problem”, Advances in Mathematics 22 (1): 64–84, doi:10.1016/0001-8708(76)90138-9, ISSN 0001-8708, MR 0437501 • Stark, Harold M. (1980), “L-functions at s = 1. IV. First derivatives at s = 0”, Advances in Mathematics 35 (3): 197–235, doi:10.1016/0001-8708(80)90049-3, ISSN 0001-8708, MR 563924 • Tate, John (1984), Les conjectures de Stark sur les fonctions L d'Artin en s=0, Progress in Mathematics 47, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3188-8, MR 782485
215.6 External links • Hayes, David R. (1999), Lectures on Stark’s Conjectures
Chapter 216
Strassmann’s theorem In mathematics, Strassmann’s theorem is a result in field theory. It states that, for suitable fields, suitable formal power series with coefficients in the valuation ring of the field have only finitely many zeroes.
216.1 History It was introduced by Reinhold Straßmann (1928).
216.2 Statement of the theorem Let K be a field with a non-Archimedean absolute value | · | and let R be the valuation ring of K. Let f(x) be a formal power series with coefficients in R other than the zero series, with coefficients an converging to zero with respect to | · |. Then f(x) has only finitely many zeroes in R. More precisely, the number of zeros is at most N, where N is the largest index with |aN| = max an.
216.3 References • Murty, M. Ram (2002). Introduction to P-Adic Analytic Number Theory. American Mathematical Society. p. 35. ISBN 978-0-8218-3262-2. • Straßmann, Reinhold (1928), "Über den Wertevorrat von Potenzreihen im Gebiet der p-adischen Zahlen.”, Journal für die reine und angewandte Mathematik (in German) 159: 13–28, doi:10.1515/crll.1928.159.13, ISSN 0075-4102, JFM 54.0162.06
216.4 External links • Weisstein, Eric W., “Strassman’s Theorem”, MathWorld.
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Chapter 217
Stufe (algebra) In field theory, the Stufe (/ʃtuːfə/; German: level) s(F) of a field F is the least number of squares that sum to −1. If −1 cannot be written as a sum of squares, s(F)= ∞ . In this case, F is a formally real field. Albrecht Pfister proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.[1]
217.1 Powers of 2 If s(F ) ̸= ∞ then s(F ) = 2k for some k ∈ N .[1][2] Proof: Let k ∈ N be chosen such that 2k ≤ s(F ) < 2k+1 . Let n = 2k . Then there are s = s(F ) elements e1 , . . . , es ∈ F \ {0} such that
0 = 1 + e21 + · · · + e2n−1 + e2n + · · · + e2s . {z } | {z } | =:a
=:b
Both a and b are sums of n squares, and a ̸= 0 , since otherwise s(F ) < 2k , contrary to the assumption on k . According to the theory of Pfister forms, the product ab is itself a sum of n squares, that is, ab = c21 + · · · + c2n for some ci ∈ F . But since a + b = 0 , we also have −a2 = ab , and hence
−1 =
( c )2 ab ( c1 )2 n = + · · · + , a2 a a
and thus s(F ) = n = 2k .
217.2 Positive characteristic The Stufe s(F ) ≤ 2 for all fields F with positive characteristic.[3] Proof: Let p = char(F ) . It suffices to prove the claim for Fp . If p = 2 then −1 = 1 = 12 , so s(F ) = 1 . If p > 2 consider the set S = {x2 | x ∈ Fp } of squares. S \ {0} is a subgroup of index 2 in the cyclic group F× p with p − 1 elements. Thus S contains exactly p+1 elements, and so does −1 − S . Since F only has p elements in p 2 total, S and −1 − S cannot be disjoint, that is, there are x, y ∈ Fp with S ∋ x2 = −1 − y 2 ∈ −1 − S and thus −1 = x2 + y 2 . 665
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217.3 Properties The Stufe s(F) is related to the Pythagoras number p(F) by p(F) ≤ s(F)+1.[4] If F is not formally real then s(F) ≤ p(F) ≤ s(F)+1.[5][6] The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F).[7][8]
217.4 Examples • The Stufe of a quadratically closed field is 1.[8] • The Stufe of an algebraic number field is ∞, 1, 2 or 4 (“Siegel’s theorem).[9] Examples are Q, Q(√−1), Q(√−2) and Q(√−7).[7] • The Stufe of a finite field GF(q) is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4.[3][8][10] • The Stufe of a local field of odd residue characteristic is equal to that of its residue field. The Stufe of the 2-adic field Q2 is 4.[9]
217.5 Notes [1] Rajwade (1993) p.13 [2] Lam (2005) p.379 [3] Rajwade (1993) p.33 [4] Rajwade (1993) p.44 [5] Rajwade (1993) p.228 [6] Lam (2005) p.395 [7] Milnor & Husemoller (1973) p.75 [8] Lam (2005) p.380 [9] Lam (2005) p.381 [10] Singh, Sahib (1974). “Stufe of a finite field”. Fibonacci Quarterly 12: 81–82. ISSN 0015-0517. Zbl 0278.12008.
217.6 References • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023. • Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. • Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
217.7 Further reading • Knebusch, Manfred; Scharlau, Winfried (1980). Algebraic theory of quadratic forms. Generic methods and Pfister forms. DMV Seminar 1. Notes taken by Heisook Lee. Boston - Basel - Stuttgart: Birkhäuser Verlag. ISBN 3-7643-1206-8. Zbl 0439.10011.
Chapter 218
Superreal number In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theory, and the study of Banach algebras. The field of superreals is itself a subfield of the surreal numbers. Dales and Woodin’s superreals are distinct from the super-real numbers of David O. Tall, which are lexicographically ordered fractions of formal power series over the reals.[1]
218.1 Formal Definition Suppose X is a Tychonoff space, also called a T₃.₅ space, and C(X) is the algebra of continuous real-valued functions on X. Suppose P is a prime ideal in C(X). Then the factor algebra A = C(X)/P is by definition an integral domain which is a real algebra and which can be seen to be totally ordered. The field of fractions F of A is a superreal field if F strictly contains the real numbers R , so that F is not order isomorphic to R . If the prime ideal P is a maximal ideal, then F is a field of hyperreal numbers (Robinson’s hyperreals being a very special case).
218.2 References [1] David Tall, “Looking at graphs through infinitesimal microscopes, windows and telescopes,” Mathematical Gazette, 64 22– 49, reprint at http://www.warwick.ac.uk/staff/David.Tall/downloads.html
218.3 Bibliography • Dales, H. Garth; Woodin, W. Hugh (1996), Super-real fields, London Mathematical Society Monographs. New Series 14, The Clarendon Press Oxford University Press, ISBN 978-0-19-853991-9, MR 1420859 • L. Gillman and M. Jerison: Rings of Continuous Functions, Van Nostrand, 1960.
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Chapter 219
Supersingular prime (for an elliptic curve) In algebraic number theory, a supersingular prime is a prime number with a certain relationship to a given elliptic curve. If the curve E defined over the rational numbers, then a prime p is supersingular for E if the reduction of E modulo p is a supersingular elliptic curve over the residue field F . Elkies (1987) showed that any elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero. Lang & Trotter (1976) conjectured that the √ X number of supersingular primes less than a bound X is within a constant multiple of ln X , using heuristics involving the distribution of Frobenius eigenvalues. As of 2012, this conjecture is open. More generally, if K is any global field—i.e., a finite extension either of Q or of F (t)—and A is an abelian variety defined over K, then a supersingular prime p for A is a finite place of K such that the reduction of A modulo p is a supersingular abelian variety.
219.1 References • Elkies, Noam D. (1987). “The existence of infinitely many supersingular primes for every elliptic curve over Q". Invent. Math. 89 (3): 561–567. doi:10.1007/BF01388985. MR 0903384. • Lang, Serge; Trotter, Hale F. (1976). Frobenius distributions in GL2 -extensions. Lecture Notes in Mathematics 504. New York: Springer-Verlag. ISBN 0-387-07550-X. Zbl 0329.12015. • Ogg, A. P. (1980). “Modular Functions”. In Cooperstein, Bruce; Mason, Geoffrey. The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif., June 25–July 20, 1979. Proc. Symp. Pure Math. 37. Providence, RI: American Mathematical Society. pp. 521–532. ISBN 0-8218-1440-0. Zbl 0448.10021. • Silverman, Joseph H. (1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106. New York: Springer-Verlag. ISBN 0-387-96203-4. Zbl 0585.14026.
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Chapter 220
Symbol (number theory) In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality. • Legendre symbol
( ) a p
defined for p a prime, a an integer, and takes values 0, 1, or −1.
( ) • Jacobi symbol ab defined for b a positive odd integer, a an integer, and takes values 0, 1, or −1. An extension of the Legendre symbol to more general values of n. ( ) • Kronecker symbol ab defined for b any integer, a an integer, and takes values 0, 1, or −1. An extension of the Jacobi and Legendre symbols to more general values of b. ( ) ( ) • Power residue symbol ab = ab m is defined for a in some global field containing the mth roots of 1 ( for some m), b a fractional ideal of K built from prime ideals coprime to m. The symbol takes values in the m roots of 1. When m = 2 and the global field is the rationals this is more or less the same as the Jacobi symbol. • Hilbert symbol The local Hilbert symbol (a,b) = is defined for a and b in some local field containing the m roots of 1 (for some m) and takes values in the m roots of 1. The ( power ) (residue ) symbol can be written in terms a,b a,b is defined for a and b in some of the Hilbert symbol. The global Hilbert symbol (a, b)p = p = p m global field K, for p a finite or infinite place of K, and is equal to the local Hilbert symbol in the completion of K at the place p. ( ) • Artin symbol The local Artin symbol or norm residue symbol θL/K (α) = (α, L/K) = L/K is defined for α L a finite extension of the local field K, α an element of K, and takes(values)in the abelianization of the Galois group Gal(L/K). The global Artin symbol ψL/K (α) = (α, L/K) = L/K = ((L/K)/α) is defined for α in α a ray class group or idele (class) group of a global field K, and takes values in the abelianization of Gal(L/K) for L an abelian extension of K. When α is in the idele group the symbol is sometimes called a Chevalley symbol or Artin–Chevalley symbol. The local Hilbert symbol of K can be written in terms of the Artin symbol for Kummer extensions L/K, where the roots of unity can be identified with elements of the Galois group. ] [ is the same as the Frobenius element of the prime P of the • The Frobenius symbol [(L/K)/P ] = L/K P Galois extension L of K. • “Chevalley symbol” has several slightly different meanings. It is sometimes used for the Artin symbol for ( ) ideles. A variation of this is the Chevalley symbol a,χ for p a prime ideal of K, a an element of K, and χ a p homomorphism of the Galois group of K to R/Z. The value of the symbol is then the value of the character χ on the usual Artin symbol. • Norm residue symbol This name is for several different closely related symbols, such as the Artin symbol ) ( or the Hilbert symbol or Hasse’s norm residue symbol. The Hasse norm residue symbol ((α, L/K)/p) = α,L/K p 669
670
CHAPTER 220. SYMBOL (NUMBER THEORY) is defined if p is a place of K and α an element of K. It is essentially the same as the local Artin symbol for the localization of K at p. The Hilbert symbol is a special case of it in the case of Kummer extensions.
• Steinberg symbol (a,b). This is a generalization of the local Hilbert symbol to arbitrary fields F. The numbers a and b are elements of F, and the symbol (a,b) takes values in the second K-group of F. • Galois symbol A sort of generalization of the Steinberg symbol to higher algebraic K-theory. It takes a Milnor K-group to an etale cohomology group.
220.1 See also • Contou-Carrère symbol • Mennicke symbol
220.2 References • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
Chapter 221
Takagi existence theorem In class field theory, the Takagi existence theorem states that for any number field K there is a one-to-one inclusion reversing correspondence between the finite abelian extensions of K (in a fixed algebraic closure of K) and the generalized ideal class groups defined via a modulus of K. It is called an existence theorem because a main burden of the proof is to show the existence of enough abelian extensions of K.
221.1 Formulation Here a modulus (or ray divisor) is a formal finite product of the valuations (also called primes or places) of K with positive integer exponents. The archimedean valuations that might appear in a modulus include only those whose completions are the real numbers (not the complex numbers); they may be identified with orderings on K and occur only to exponent one. The modulus m is a product of a non-archimedean (finite) part mf and an archimedean (infinite) part m∞. The non-archimedean part mf is a nonzero ideal in the ring of integers OK of K and the archimedean part m∞ is simply a set of real embeddings of K. Associated to such a modulus m are two groups of fractional ideals. The larger one, Im, is the group of all fractional ideals relatively prime to m (which means these fractional ideals do not involve any prime ideal appearing in mf). The smaller one, Pm, is the group of principal fractional ideals (u/v) where u and v are nonzero elements of OK which are prime to mf, u ≡ v mod mf, and u/v > 0 in each of the orderings of m∞. (It is important here that in Pm, all we require is that some generator of the ideal has the indicated form. If one does, others might not. For instance, taking K to be the rational numbers, the ideal (3) lies in P 4 because (3) = (−3) and −3 fits the necessary conditions. But (3) is not in P₄∞ since here it is required that the positive generator of the ideal is 1 mod 4, which is not so.) For any group H lying between Im and Pm, the quotient Im/H is called a generalized ideal class group. It is these generalized ideal class groups which correspond to abelian extensions of K by the existence theorem, and in fact are the Galois groups of these extensions. That generalized ideal class groups are finite is proved along the same lines of the proof that the usual ideal class group is finite, well in advance of knowing these are Galois groups of finite abelian extensions of the number field.
221.2 A well-defined correspondence Strictly speaking, the correspondence between finite abelian extensions of K and generalized ideal class groups is not quite one-to-one. Generalized ideal class groups defined relative to different moduli can give rise to the same abelian extension of K, and this is codified a priori in a somewhat complicated equivalence relation on generalized ideal class groups. In concrete terms, for abelian extensions L of the rational numbers, this corresponds to the fact that an abelian extension of the rationals lying in one cyclotomic field also lies in infinitely many other cyclotomic fields, and for each such cyclotomic overfield one obtains by Galois theory a subgroup of the Galois group corresponding to the same 671
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field L. In the idelic formulation of class field theory, one obtains a precise one-to-one correspondence between abelian extensions and appropriate groups of ideles, where equivalent generalized ideal class groups in the ideal-theoretic language correspond to the same group of ideles.
221.3 Earlier work A special case of the existence theorem is when m = 1 and H = P 1 . In this case the generalized ideal class group is the ideal class group of K, and the existence theorem says there exists a unique abelian extension L/K with Galois group isomorphic to the ideal class group of K such that L is unramified at all places of K. This extension is called the Hilbert class field. It was conjectured by David Hilbert to exist, and existence in this special case was proved by Furtwängler in 1907, before Takagi’s general existence theorem. A further and special property of the Hilbert class field, not true of smaller abelian extensions of a number field, is that all ideals in a number field become principal in the Hilbert class field. It required Artin and Furtwängler to prove that principalization occurs.
221.4 History The existence theorem is due to Takagi, who proved it in Japan during the isolated years of World War I. He presented it at the International Congress of Mathematicians in 1920, leading to the development of the classical theory of class field theory during the 1920s. At Hilbert’s request, the paper was published in Mathematische Annalen in 1925.
221.5 See also • Class formation
221.6 References • Helmut Hasse, History of Class Field Theory, pp. 266–279 in Algebraic Number Theory, eds. J. W. S. Cassels and A. Fröhlich, Academic Press 1967. (See also the rich bibliography attached to Hasse’s article.)
Chapter 222
Tate cohomology group In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by John Tate (1952, p. 297), and are used in class field theory.
222.1 Definition If G is a finite group and A a G-module, then there is a natural map N from H 0 (G,A) to H 0 (G,A) taking a representative ˆ n (G, A) are defined by a to Σ g(a) (the sum over all G-conjugates of a). The Tate cohomology groups H ˆ n (G, A) = H n (G, A) for n≥ 1. • H ˆ 0 (G, A) = quotient of H 0 (G,A) by norms • H ˆ −1 (G, A) = quotient of norm 0 elements of H 0 (G,A) by principal norm 0 elements • H ˆ n (G, A) = H−(n+1) (G, A) for n≤ −2. • H
222.2 Properties If
0 −→ A −→ B −→ C −→ 0 is a short exact sequence of G-modules, then we get the usual long exact sequence of Tate cohomology groups:
ˆ n (G, A) −→ H ˆ n (G, B) −→ H ˆ n (G, C) −→ H ˆ n+1 (G, A) −→ H ˆ n+1 (G, B) · · · · · · −→ H If A is an induced G module then all Tate cohomology groups of A vanish. The zeroth Tate cohomology group of A is (Fixed points of G on A)/(Obvious fixed points of G acting on A) where by the “obvious” fixed point we mean those of the form Σ g(a). In other words, the zeroth cohomology group in some sense describes the non-obvious fixed points of G acting on A. The Tate cohomology groups are characterized by the three properties above. 673
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222.3 Tate’s theorem Tate’s theorem (Tate 1952) gives conditions for multiplication by a cohomology class to be an isomorphism between cohomology groups. There are several slightly different versions of it; a version that is particularly convenient for class field theory is as follows: Suppose that A is a module over a finite group G and a is an element of H 2 (G,A), such that for every subgroup E of G • H 1 (E,A) is trivial, and • H 2 (E,A) is generated by Res(a) which has order E. Then cup product with a is an isomorphism ˆ n (G, Z) −→ H ˆ n+2 (G, A) • H for all n; in other words the graded Tate cohomology of A is isomorphic to the Tate cohomology with integral coefficients, with the degree shifted by 2.
222.4 Tate-Farrell cohomology Farrell extended Tate cohomology groups to the case of all groups G of finite virtual cohomological dimension. In ˆ n (G, A) are isomorphic to the usual cohomology groups whenever n is greater than the Farrell’s theory, the groups H virtual cohomological dimension of the group G. Finite groups have virtual cohomological dimension 0, and in this case Farrell’s cohomology groups are the same as those of Tate.
222.5 See also • Herbrand quotient • Class formation
222.6 References • M. F. Atiyah and C. T. C. Wall, “Cohomology of Groups”, in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2, Chapter IV. See section 6. • Kenneth S. Brown, Cohomology of Groups, ISBN 0-387-90688-6 • Farrell, F. Thomas An extension of Tate cohomology to a class of infinite groups. J. Pure Appl. Algebra 10 (1977/78), no. 2, 153-161. • Tate, John (1952), “The higher dimensional cohomology groups of class field theory”, Ann. of Math. (2) 56: 294–297, doi:10.2307/1969801
Chapter 223
Tate duality In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by Tate (1962) and Poitou (1967).
223.1 Local Tate duality Main article: local Tate duality Local Tate duality says there is a perfect pairing of finite groups
H r (k, M ) × H 2−r (k, M ′ ) → H 2 (k, Gm ) = Q/Z where M is a finite group scheme and M′ its dual Hom(M,Gm).
223.2 See also • Artin–Verdier duality • Tate pairing
223.3 References • Haberland, Klaus (1978), Galois cohomology of algebraic number fields, VEB Deutscher Verlag der Wissenschaften, MR 519872 • Poitou, Georges (1967), “Propriétés globales des modules finis”, Cohomologie galoisienne des modules finis, Séminaire de l'Institut de Mathématiques de Lille, sous la direction de G. Poitou. Travaux et Recherches Mathématiques, 13, Paris: Dunod, pp. 255–277, MR 0219591 • Tate, John (1963), “Duality theorems in Galois cohomology over number fields”, Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 288–295, MR 0175892
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Tate’s thesis In number theory, Tate’s thesis is the 1950 thesis of John Tate (1950) under supervision of Emil Artin. In it, he used a translation invariant integration on the locally compact group of ideles to lift the zeta function of a number field, twisted by a Hecke character, to a zeta integral and study its properties. Using harmonic analysis, more precisely the summation formula, he proved the functional equation and meromorphic continuation of the zeta integral and the twisted zeta function. He also located the poles of the twisted zeta function. His work can be viewed as an elegant and powerful reformulation of a work of Erich Hecke on the proof of the functional equation of the twisted zeta function (L-function). Hecke used a generalized theta series associated to an algebraic number field and a lattice in its ring of integers. Kenkichi Iwasawa independently discovered during the war essentially the same method (without an analog of the local theory in Tate’s thesis) and announced it in his 1950 ICM paper and his letter to Dieudonné written in 1952. Hence this theory is often called Iwasawa–Tate theory. Iwasawa in his letter to Dieudonné derived on several pages not only the meromorphic continuation and functional equation of the L-function, he also proved finiteness of the class number and Dirichlet’s theorem on units as immediate byproducts of the main computation. The theory in positive characteristic was developed one decade earlier by Witt, Schmid and Teichmuller. Iwasawa-Tate theory uses several structures which come from class field theory, however it does not use any deep result of class field theory.
224.1 Generalisations A noncommutative generalisation: Iwasawa-Tate theory was extended to a general linear group over an algebraic number field and automorphic representations of its adelic group by Roger Godement and Hervé Jacquet in 1972. This work is part of activities in the Langlands correspondence. A higher-dimensional generalisation: unramified Iwasawa-Tate theory was extended to a regular model of an elliptic curve over an algebraic number field and the function field of a curve over a finite field by Ivan Fesenko in 2010. This work is part of activities in the study of the arithmetic zeta functions of arithmetic schemes using complex analytic and higher adelic methods. It uses K-theoretical structures which are involved in higher class field theory but does not use deep results of the latter.
224.2 References • Fesenko, Ivan (2010), “Analysis on arithmetic schemes. II”, J. K-theory (Cambridge University Press) 5: 437– 557 • Godement, Roger; Jacquet, Hervé (1972), Zeta functions of simple algebras, Lect. Notes Math. 260, Springer • Goldfeld, Dorian; Hundley, Joseph (2011), Automorphic representations of L-functions for the generali linear group, Cambridge University Press • Iwasawa, Kenkichi (1952), “A note on functions”, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950 1, Providence, R.I.: American Mathematical Society, p. 322, MR 0044534 676
224.2. REFERENCES
677
• Iwasawa, Kenkichi (1992) [1952], “Letter to J. Dieudonné", in Kurokawa, Nobushige; Sunada., T., Zeta functions in geometry (Tokyo, 1990), Adv. Stud. Pure Math. 21, Tokyo: Kinokuniya, pp. 445–450, ISBN 978-4314-10078-6, MR 1210798 • Kudla, Stephen S. (2003), “Tate’s thesis”, in Bernstein, Joseph; Gelbart, Stephen, An introduction to the Langlands program (Jerusalem, 2001), Boston, MA: Birkhäuser Boston, pp. 109–131, ISBN 978-0-8176-3211-3, MR 1990377 • Ramakrishnan, Dinakar; Valenza, Robert J. (1999). Fourier analysis on number fields. Graduate Texts in Mathematics 186. New York: Springer-Verlag. doi:10.1007/978-1-4757-3085-2. ISBN 0-387-98436-4. MR 1680912. • Tate, John T. (1950), “Fourier analysis in number fields, and Hecke’s zeta-functions”, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-09502734-2-6, MR 0217026
Chapter 225
Teichmüller character In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)× , where q = p if p is odd and q=4 if p = 2, taking values in the roots of unity of the p-adic integers. It was introduced by Oswald Teichmüller. Identifying the roots of unity in the p-adic integers with the corresponding ones in the complex numbers, ω can be considered as a usual Dirichlet character of conductor q. More generally, given a complete discrete valuation ring O whose residue field k is perfect of characteristic p, there is a unique multiplicative section ω : k → O of the natural surjection O → k. The image of an element under this map is called its Teichmüller representative. The restriction of ω to k× is called the Teichmüller character.
225.1 Definition If x is a p-adic integer, then ω(x) is the unique solution of ω(x)p = ω(x) that is congruent to x mod p. It can also be defined by
ω(x) = lim xp
n
n→∞
The multiplicative group of p-adic units is a product of the finite group of roots of unity and a group isomorphic to the p-adic integers. The finite group is cyclic of order p – 1 or 2, as p is odd or even, respectively, and so it is isomorphic to (Z/qZ)× . The Teichmüller character gives a canonical isomorphism between these two groups.
225.2 References • Section 4.3 of Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337 • Koblitz, Neal (1984), p-adic Numbers, p-adic Analysis, and Zeta-Functions, Graduate Texts in Mathematics, vol. 58, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96017-3, MR 754003
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Chapter 226
Teichmüller cocycle In mathematics, the Teichmüller cocycle is a certain 3-cocycle associated to a simple algebra A over a field L which is a finite Galois extension of a field K and which has the property that any automorphism of L over K extends to an automorphism of A. The Teichmüller cocycle, or rather its cohomology class, is the obstruction to the algebra A coming from a simple algebra over K. It was introduced by Teichmüller (1940) and named by Eilenberg and MacLane (1948).
226.1 Properties If K is a finite normal extension of the global field k, then the Galois cohomology group H3 (Gal(K/k,K*) is cyclic and generated by the Teichmüller cocycle. Its order is n/m where n is the degree of the extension K/k and m is the least common multiple of all the local degrees (Artin & Tate 2009, p.68).
226.2 References • Artin, Emil; Tate, John (2009) [1952], Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, MR 0223335 • Eilenberg, Samuel; MacLane, Saunders (1948), “Cohomology and Galois theory. I. Normality of algebras and Teichmüller’s cocycle.”, Trans. Amer. Math. Soc. 64: 1–20, doi:10.1090/s0002-9947-1948-0025443-3, MR 0025443 • Teichmüller, Oswald (1940), "Über die sogenannte nichtkommutative Galoissche Theorie und die Relation λ ξλ,µ,ν ξλ,µν,π ξµ,ν,π = ξλ,µ,νπ ξλ,µ,ν,π ", Deutsche Mathematik: 138–149
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Chapter 227
Tensor product of fields In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. On the other hand it is often required to 'join' two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K and L are both field extensions of a smaller field N (for example a prime field). The tensor product of fields is the best available construction on fields with which to discuss all the phenomena arising. As a ring, it is sometimes a field, and often a direct product of fields; it can, though, contain non-zero nilpotents (see radical of a ring). If K and L do not have isomorphic prime fields, or in other words they have different characteristics, they have no possibility of being common subfields of a field M. Correspondingly their tensor product will in that case be the trivial ring (collapse of the construction to nothing of interest).
227.1 Compositum of fields Firstly, one defines the notion of the compositum of fields. This construction occurs frequently in field theory. The idea behind the compositum is to make the smallest field containing two other fields. In order to formally define the compositum, one must first specify a tower of fields. Let k be a field and L and K be two extensions of k. The compositum, denoted KL is defined to be KL = k(K ∪ L) where the right-hand side denotes the extension generated by K and L. Note that this assumes some field containing both K and L. Either one starts in a situation where such a common over-field is easy to identify (for example if K and L are both subfields of the complex numbers); or one proves a result that allows one to place both K and L (as isomorphic copies) in some large enough field. In many cases one can identify K.L as a vector space tensor product, taken over the field N that is the intersection of K and L. For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M obtained as K.L inside the complex numbers ℂ is (up to isomorphism)
K ⊗Q L as a vector space over ℚ. (This type of result can be verified, in general, by using the ramification theory of algebraic number theory.) Subfields K and L of M are linearly disjoint (over a subfield N) when in this way the natural N-linear map of
K ⊗N L to K.L is injective.[1] Naturally enough this isn't always the case, for example when K = L. When the degrees are finite, injective is equivalent here to bijective. A significant case in the theory of cyclotomic fields is that for the nth roots of unity, for n a composite number, the subfields generated by the pk th roots of unity for prime powers dividing n are linearly disjoint for distinct p.[2] 680
227.2. THE TENSOR PRODUCT AS RING
681
227.2 The tensor product as ring To get a general theory, one needs to consider a ring structure on K ⊗N L . One can define the product (a ⊗ b)(c ⊗ d) to be ac⊗bd . This formula is multilinear over N in each variable; and so defines a ring structure on the tensor product, making K ⊗N L into a commutative N-algebra, called the tensor product of fields.
227.3 Analysis of the ring structure The structure of the ring can be analysed by considering all ways of embedding both K and L in some field extension of N. Note that the construction here assumes the common subfield N; but does not assume a priori that K and L are subfields of some field M (thus getting round the caveats about constructing a compositum field). Whenever one embeds K and L in such a field M, say using embeddings α of K and β of L, there results a ring homomorphism γ from K ⊗N L into M defined by:
γ(a ⊗ b) = (α(a) ⊗ 1) ⋆ (1 ⊗ β(b)) = α(a).β(b). The kernel of γ will be a prime ideal of the tensor product; and conversely any prime ideal of the tensor product will give a homomorphism of N-algebras to an integral domain (inside a field of fractions) and so provides embeddings of K and L in some field as extensions of (a copy of) N. In this way one can analyse the structure of K ⊗N L : there may in principle be a non-zero Jacobson radical (intersection of all prime ideals) - and after taking the quotient by that one can speak of the product of all embeddings of K and L in various M, over N. In case K and L are finite extensions of N, the situation is particularly simple since the tensor product is of finite dimension as an N-algebra (and thus an Artinian ring). One can then say that if R is the radical, one has (K ⊗N L)/R as a direct product of finitely many fields. Each such field is a representative of an equivalence class of (essentially distinct) field embeddings for K and L in some extension M.
227.4 Examples For example, if K is generated over ℚ by the cube root of 2, then K ⊗Q K is the product of (a copy of) K, and a splitting field of X3 − 2, of degree 6 over ℚ. One can prove this by calculating the dimension of the tensor product over ℚ as 9, and observing that the splitting field does contain two (indeed three) copies of K, and is the compositum of two of them. That incidentally shows that R = {0} in this case. An example leading to a non-zero nilpotent: let P(X) = Xp − T with K the field of rational functions in the indeterminate T over the finite field with p elements. (See separable polynomial: the point here is that P is not separable). If L is the field extension K(T 1/p ) (the splitting field of P) then L/K is an example of a purely inseparable field extension. In L ⊗K L the element
T 1/p ⊗ 1 − 1 ⊗ T 1/p is nilpotent: by taking its pth power one gets 0 by using K-linearity.
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227.5 Classical theory of real and complex embeddings In algebraic number theory, tensor products of fields are (implicitly, often) a basic tool. If K is an extension of ℚ of finite degree n, K ⊗Q R is always a product of fields isomorphic to ℝ or ℂ. The totally real number fields are those for which only real fields occur: in general there are r1 real and r2 complex fields, with r1 + 2r2 = n as one sees by counting dimensions. The field factors are in 1–1 correspondence with the real embeddings, and pairs of complex conjugate embeddings, described in the classical literature. This idea applies also to K ⊗Q Qp , where ℚp is the field of p-adic numbers. This is a product of finite extensions of ℚp, in 1–1 correspondence with the completions of K for extensions of the p-adic metric on ℚ.
227.6 Consequences for Galois theory This gives a general picture, and indeed a way of developing Galois theory (along lines exploited in Grothendieck’s Galois theory). It can be shown that for separable extensions the radical is always {0}; therefore the Galois theory case is the semisimple one, of products of fields alone.
227.7 See also • Extension of scalars—tensor product of a field extension and a vector space over that field
227.8 Notes [1] Hazewinkel, Michiel, ed. (2001), “Linearly-disjoint extensions”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4 [2] Hazewinkel, Michiel, ed. (2001), “Cyclotomic field”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
227.9 References • Hazewinkel, Michiel, ed. (2001), “Compositum of field extensions”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • George Kempf (1995) Algebraic Structures, pp. 85–87. • Algebraic Number Theory, J. S. Milne Notes (PDF) at p. 17. • A Brief Introduction to Classical and Adelic Algebraic Number Theory, William Stein (PDF) pp. 140–142. • Zariski, Oscar; Samuel, Pierre (1975) [1958], Commutative algebra I, Graduate Texts in Mathematics 28, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90089-6, MR 0090581
227.10 External links • MathOverflow thread on the definition of linear disjointness
Chapter 228
Thin set (Serre) For other uses, see Thin set. In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within K a polynomial that does not always factorise. One is also allowed to take finite unions.
228.1 Formulation More precisely, let V be an algebraic variety over K (assumptions here are: V is an irreducible set, a quasi-projective variety, and K has characteristic zero). A type I thin set is a subset of V(K) that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than d, the dimension of V. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the K-points of some other d-dimensional algebraic variety V′, that maps essentially onto V as a ramified covering with degree e > 1. Saying this more technically, a thin set of type II is any subset of φ(V′(K)) where V′ satisfies the same assumptions as V and φ is generically surjective from the geometer’s point of view. At the level of function fields we therefore have [K(V): K(V′)] = e > 1. While a typical point v of V is φ(u) with u in V′, from v lying in K(V) we can conclude typically only that the coordinates of u come from solving a degree e equation over K. The whole object of the theory of thin sets is then to understand that the solubility in question is a rare event. This reformulates in more geometric terms the classical Hilbert irreducibility theorem. A thin set, in general, is a subset of a finite union of thin sets of types I and II . The terminology thin may be justified by the fact that if A is a thin subset of the line over Q(then the ) number of points of A of height at most H is ≪ H: the number of integral points of height at most H is O N 1/2 , and this result is best possible.[1] A result of S. D. Cohen, based on the large sieve method, extends this result, counting points by height function and showing, in a strong sense, that a thin set contains a low proportion of them (this is discussed at length in Serre’s Lectures on the Mordell-Weil theorem). Let A be a thin set in affine n-space over Q and let N(H) denote the number of integral points of naive height at most H. Then[2] ( ) N (H) = O H n−1/2 log H . 683
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228.2 Hilbertian fields A Hilbertian variety V over K is one for which V(K) is not thin: this is a birational invariant of V.[3] A Hilbertian field K is one for which there exists a Hilbertian variety of positive dimension over K:[3] the term was introduced by Lang in 1962.[4] If K is Hilbertian then the projective line over K is Hilbertian, so this may be taken as the definition.[5][6] The rational number field Q is Hilbertian, because Hilbert’s irreducibility theorem has as a corollary that the projective line over Q is Hilbertian: indeed, any algebraic number field is Hilbertian, again by the Hilbert irreducibility theorem.[5][7] More generally a finite degree extension of a Hilbertian field is Hilbertian[8] and any finitely generated infinite field is Hilbertian.[6] There are several results on the permanence criteria of Hilbertian fields. Notably Hilbertianity is preserved under finite separable extensions[9] and abelian extensions. If N is a Galois extension of a Hilbertian field, then although N need not be Hilbertian itself, Weisseauer’s results asserts that any proper finite extension of N is Hilbertian. The most general result in this direction is Haran’s diamond theorem. A discussion on these results and more appears in Fried-Jarden’s Field Arithmetic. Being Hilbertian is at the other end of the scale from being algebraically closed: the complex numbers have all sets thin, for example. They, with the other local fields (real numbers, p-adic numbers) are not Hilbertian.[5]
228.3 WWA property The WWA property (weak 'weak approximation', sic) for a variety V over a number field is weak approximation (cf. approximation in algebraic groups), for finite sets of places of K avoiding some given finite set. For example take K = Q: it is required that V(Q) be dense in Π V(Qp) for all products over finite sets of prime numbers p, not including any of some set {p1 , ..., pM} given once and for all. Ekedahl has proved that WWA for V implies V is Hilbertian.[10] In fact Colliot-Thélène conjectures WWA holds for any unirational variety, which is therefore a stronger statement. This conjecture would imply a positive answer to the inverse Galois problem.[10]
228.4 References [1] Serre (1992) p.26 [2] Serre (1992) p.27 [3] Serre (1992) p.19 [4] Schinzel (2000) p.312 [5] Serre (1992) p.20 [6] Schinzel (2000) p.298 [7] Lang (1997) p.41 [8] Serre (1992) p.21 [9] Fried & Jarden (2008) p.224 [10] Serre (1992) p.29
• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001. • Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
228.4. REFERENCES
685
• Serre, Jean-Pierre (1989). Lectures on the Mordell-Weil Theorem. Aspects of Mathematics E15. Translated and edited by Martin Brown from notes by Michel Waldschmidt. Braunschweig etc.: Friedr. Vieweg & Sohn. Zbl 0676.14005. • Serre, Jean-Pierre (1992). Topics in Galois Theory. Research Notes in Mathematics 1. Jones and Bartlett. ISBN 0-86720-210-6. Zbl 0746.12001. • Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001.
Chapter 229
Timeline of class field theory In mathematics, class field theory is the study of abelian extensions of local and global fields.
229.1 Timeline • 1801 Gauss proves the law of quadratic reciprocity • 1829 Abel uses special values of the lemniscate function to construct abelian extensions of Q(i). • 1837 Dirichlet’s theorem on arithmetic progressions. • 1853 Kronecker announces the Kronecker-Weber theorem • 1880 Kronecker introduces his Jugendtraum about abelian extensions of imaginary quadratic fields • 1886 Weber proves the Kronecker-Weber theorem (with a slight gap) • 1896 Hilbert gives the first complete proof of the Kronecker-Weber theorem • 1897 Weber introduces ray class groups and general ideal class groups • 1897 Hilbert publishes his Zahlbericht. • 1897 Hilbert rewrites the law of quadratic reciprocity as a product formula for the Hilbert symbol. • 1897 Hensel introduced p-adic numbers • 1898 Hilbert conjectures the existence and properties of the (narrow) Hilbert class field, proving them in the special case of class number 2. • 1907 Furtwangler proves existence and basic properties of the Hilbert class field • 1908 Weber defines the class field of a general ideal class group • 1920 Takagi shows that the abelian extensions of a number field are exactly the class fields of ideal class groups. • 1922 Takagi’s paper on reciprocity laws • 1923 Hasse introduced the Hasse principle (for the special case of quadratic forms). • 1923 Artin conjectures his reciprocity law • 1924 Artin introduces Artin L-functions • 1926 Chebotarev proves his density theorem • 1927 Artin proves his reciprocity law giving a canonical isomorphism between Galois groups and ideal class groups 686
229.2. REFERENCES
687
• 1930 Furtwangler and Artin prove the principal ideal theorem • 1930 Hasse introduces local class field theory • 1931 Hasse proves the Hasse norm theorem • 1931 Hasse classifies simple algebras over local fields • 1931 Herbrand introduces the Herbrand quotient. • 1931 The Brauer-Hasse-Noether theorem proves the Hasse principle for simple algebras over global fields. • 1933 Hasse classifies simple algebras over number fields • 1934 Deuring and Noether develop class field theory using algebras • 1936 Chevalley introduces ideles • 1940 Chevalley uses ideles to give an algebraic proof of the second inequality for abelian extensions • 1948 Wang proves the Grunwald–Wang theorem, correcting an error of Grunwald’s. • 1950 Tate’s thesis uses analysis on adele rings to study zeta functions • 1951 Weil introduces Weil groups • 1952 Artin and Tate introduce class formations in their notes on class field theory • 1952 Hochschild and Nakayama introduce group cohomology into class field theory • 1952 Tate introduces Tate cohomology groups • 1964 Golod and Shafarevich prove that the class field tower can be infinite • 1965 Lubin and Tate use Lubin–Tate formal group laws to construct ramified abelian extensions of local fields.
229.2 References • Conrad, Keith, History of class field theory • Hasse, Helmut (1967), “History of class field theory”, Algebraic Number Theory, Washington, D.C.: Thompson, pp. 266–279, MR 0218330 • Iyanaga, S. (1975) [1969], “History of class field theory”, The theory of numbers, Noth Holland, pp. 479–518 • Roquette, Peter (2001), “Class field theory in characteristic p, its origin and development”, Class field theory— its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math. 30, Tokyo: Math. Soc. Japan, pp. 549–631,
Chapter 230
Totally imaginary number field In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotomic fields, and, more generally, CM fields. Any number field that is Galois over the rationals must be either totally real or totally imaginary.
230.1 References • Section 13.1 of Alaca, Şaban; Williams, Kenneth S. (2004), Introductory algebraic number theory, Cambridge University Press, ISBN 978-0-521-54011-7
688
Chapter 231
Totally real number field
The number field Q(√2) sits inside R, and the two embeddings of the field into C send every element in the field to another element of R, hence the field is totally real.
In number theory, a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers. Equivalent conditions are that K is generated over Q by one root of an integer 689
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CHAPTER 231. TOTALLY REAL NUMBER FIELD
polynomial P, all of the roots of P being real; or that the tensor product algebra of K with the real field, over Q, is a product of copies of R. For example, quadratic fields K of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial P irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will not be totally real, although it is a field of real numbers. The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally real, or contains a totally real subfield over which it has degree two. Any number field that is Galois over the rationals must be either totally real or totally imaginary.
231.1 See also • Totally imaginary number field • CM-field, a totally imaginary quadratic extension of a totally real field.
231.2 References • Hida, Haruzo (1993), Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts 26, Cambridge University Press, ISBN 978-0-521-43569-7
Chapter 232
Tower of fields In mathematics, a tower of fields is a sequence of field extensions F 0 ⊆ F 1 ⊆ ... ⊆ Fn ⊆ ... The name comes from such sequences often being written in the form .. . | F2 | F1 | F0 . A tower of fields may be finite or infinite.
232.1 Examples • Q ⊆ R ⊆ C is a finite tower with rational, real and complex numbers. • The sequence obtained by letting F 0 be the rational numbers Q, and letting ( ) n Fn+1 = Fn 21/2 (i.e. Fn₊₁ is obtained from Fn by adjoining a 2n th root of 2) is an infinite tower. • If p is a prime number the p th cyclotomic tower of Q is obtained by letting F 0 = Q and Fn be the field obtained by adjoining to Q the pn th roots of unity. This tower is of fundamental importance in Iwasawa theory. • The Golod–Shafarevich theorem shows that there are infinite towers obtained by iterating the Hilbert class field construction to a number field.
232.2 References • Section 4.1.4 of Escofier, Jean-Pierre (2001), Galois theory, Graduate Texts in Mathematics 204, SpringerVerlag, ISBN 978-0-387-98765-1
691
Chapter 233
Transcendence degree In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the “size” of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of L over K. A subset S of L is a transcendence basis of L /K if it is algebraically independent over K and if furthermore L is an algebraic extension of the field K(S) (the field obtained by adjoining the elements of S to K). One can show that every field extension has a transcendence basis, and that all transcendence bases have the same cardinality; this cardinality is equal to the transcendence degree of the extension and is denoted trdegK L or trdeg(L /K). If no field K is specified, the transcendence degree of a field L is its degree relative to the prime field of the same characteristic, i.e., Q if L is of characteristic 0 and Fp if L is of characteristic p. The field extension L /K is purely transcendental if there is a subset S of L that is algebraically independent over K and such that L = K(S).
233.1 Examples • An extension is algebraic if and only if its transcendence degree is 0; the empty set serves as a transcendence basis here. • The field of rational functions in n variables K(x1 ,...,xn) is a purely transcendental extension with transcendence degree n over K; we can for example take {x1 ,...,xn} as a transcendence base. • More generally, the transcendence degree of the function field L of an n-dimensional algebraic variety over a ground field K is n. • Q(√2, π) has transcendence degree 1 over Q because √2 is algebraic while π is transcendental. • The transcendence degree of C or R over Q is the cardinality of the continuum. (This follows since any element has only countably many algebraic elements over it in Q, since Q is itself countable.) • The transcendence degree of Q(π, e) over Q is either 1 or 2; the precise answer is unknown because it is not known whether π and e are algebraically independent.
233.2 Analogy with vector space dimensions There is an analogy with the theory of vector space dimensions. The dictionary matches algebraically independent sets with linearly independent sets; sets S such that L is algebraic over K(S) with spanning sets; transcendence bases with bases; and transcendence degree with dimension. The fact that transcendence bases always exist (like the fact that bases always exist in linear algebra) requires the axiom of choice. The proof that any two bases have the same cardinality depends, in each setting, on an exchange lemma.[1] This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of matroids, called linear matroids and algebraic matroids respectively. 692
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693
Thus, the transcendence degree is the rank function of an algebraic matroid. Every linear matroid is isomorphic to an algebraic matroid, but not vice versa.[2]
233.3 Facts If M/L is a field extension and L /K is another field extension, then the transcendence degree of M/K is equal to the sum of the transcendence degrees of M/L and L/K. This is proven by showing that a transcendence basis of M/K can be obtained by taking the union of a transcendence basis of M/L and one of L /K.
233.4 Applications Transcendence bases are a useful tool to prove various existence statements about field homomorphisms. Here is an example: Given an algebraically closed field L, a subfield K and a field automorphism f of K, there exists a field automorphism of L which extends f (i.e. whose restriction to K is f). For the proof, one starts with a transcendence basis S of L/K. The elements of K(S) are just quotients of polynomials in elements of S with coefficients in K; therefore the automorphism f can be extended to one of K(S) by sending every element of S to itself. The field L is the algebraic closure of K(S) and algebraic closures are unique up to isomorphism; this means that the automorphism can be further extended from K(S) to L. As another application, we show that there are (many) proper subfields of the complex number field C which are (as fields) isomorphic to C. For the proof, take a transcendence basis S of C/Q. S is an infinite (even uncountable) set, so there exist (many) maps f: S → S which are injective but not surjective. Any such map can be extended to a field homomorphism Q(S) → Q(S) which is not surjective. Such a field homomorphism can in turn be extended to the algebraic closure C, and the resulting field homomorphisms C → C are not surjective. The transcendence degree can give an intuitive understanding of the size of a field. For instance, a theorem due to Siegel states that if X is a compact, connected, complex manifold of dimension n and K(X) denotes the field of (globally defined) meromorphic functions on it, then trdegC(K(X)) ≤ n.
233.5 References [1] J.S. Milne, Fields and Galois Theory, pp.100-101. [2] Joshi, K. D. (1997), Applied Discrete Structures, New Age International, p. 909, ISBN 9788122408263.
Chapter 234
Tschirnhaus transformation In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. It may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root. In detail, let K be a field, and P(t) a polynomial over K. If P is irreducible, then K[t]/(P(t)) = L, the quotient ring of the polynomial ring K[t] by the principal ideal generated by P, is a field extension of K. We have L = K(α) where α is t modulo (P). That is, α is a primitive element of L. There will be other choices β of primitive element in L: for any such choice of β we will have β = F(α), α = G(β), with polynomials F and G over K. In fact this follows from the quotient representation above. Now if Q is the minimal polynomial for β over K, we can call Q a Tschirnhaus transformation of P. Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing P, but leaving L the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory, when L is a Galois extension of K. The Galois group is then described (in one way) as all the Tschirnhaus transformations of P to itself.
234.1 See also • Polynomial transformations
234.2 References • Weisstein, Eric W., “Tschirnhausen Transformation”, MathWorld. • http://www.sigsam.org/bulletin/articles/143/tschirnhaus.pdf A translation (by RF Green) of his 1683 paper— A method for removing all intermediate terms from a given equation.
694
Chapter 235
Tsen rank In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their study in 1936. We consider a system of m polynomial equations in n variables over a field F. Assume that the equations all have constant term zero, so that (0, 0, ... ,0) is a common solution. We say that F is a Ti-field if every such system, of degrees d1 , ..., dm has a common non-zero solution whenever n > di1 + · · · + dim . The Tsen rank of F is the smallest i such that F is a Ti-field. We say that the Tsen rank of F is infinite if it is not a Ti-field for any i (for example, if it is formally real).
235.1 Properties • A field has Tsen rank zero if and only if it is algebraically closed. • A finite field has Tsen rank 1: this is the Chevalley–Warning theorem. • If F is algebraically closed then rational function field F(X) has Tsen rank 1. • If F has Tsen rank i, then the rational function field F(X) has Tsen rank at most i + 1. • If F has Tsen rank i, then an algebraic extension of F has Tsen rank at most i. • If F has Tsen rank i, then an extension of F of transcendence degree k has Tsen rank at most i + k. • There exist fields of Tsen rank i for every integer i ≥ 0.
235.2 Norm form We define a norm form of level i on a field F to be a homogeneous polynomial of degree d in n=di variables with only the trivial zero over F (we exclude the case n=d=1). The existence of a norm form on level i on F implies that F is of Tsen rank at least i − 1. If E is an extension of F of finite degree n > 1, then the field norm form for E/F is a norm form of level 1. If F admits a norm form of level i then the rational function field F(X) admits a norm form of level i + 1. This allows us to demonstrate the existence of fields of any given Tsen rank.
235.3 Diophantine dimension The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class Ck: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > dk . Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.[1] 695
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Clearly if a field is Ti then it is Ci, and T0 and C0 are equivalent, each being equivalent to being algebraically closed. It is not known whether Tsen rank and Diophantine dimension are equal in general.
235.4 See also • Tsen’s theorem
235.5 References [1] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 3-540-37888-X.
• Tsen, C. (1936). “Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper”. J. Chinese Math. Soc. 171: 81–92. Zbl 0015.38803. • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4.
Chapter 236
Twisted polynomial ring In mathematics, a twisted polynomial is a polynomial over a field of characteristic p in the variable τ representing the Frobenius map x 7→ xp . In contrast to normal polynomials, multiplication of these polynomials is not commutative, but satisfies the commutation rule
τ x = xp τ for all x . Over an infinite field, the twisted polynomial ring is isomorphic to the ring of additive polynomials, but where multiplication on the latter is given by composition rather than usual multiplication. However, it is often easier to compute in the twisted polynomial ring — this can be applied especially in the theory of Drinfeld modules.
236.1 Definition Let k be a field of characteristic p . The twisted polynomial ring k{τ } is defined as the set of polynomials in the variable τ and coefficients in k . It is endowed with a ring structure with the usual addition, but with a non-commutative multiplication that can be summarized with the relation τ x = xp τ . Repeated application of this relation yields a formula for the multiplication of any two twisted polynomials. As an example we perform such a multiplication
(a + bτ )(c + dτ ) = a(c + dτ ) + bτ (c + dτ ) = ac + adτ + bcp τ + bdp τ 2
236.2 Properties The morphism
k{τ } → k[x],
a0 + a1 τ + · · · + an τ n 7→ a0 x + a1 xp + · · · + an xp
n
defines a ring homomorphism sending a twisted polynomial to an additive polynomial. Here, multiplication on the right hand side is given by composition of polynomials. For example
2
(ax + bxp ) ◦ (cx + dxp ) = a(cx + dxp ) + b(cx + dxp )p = acx + adxp + bcp xp + bdp xp , using the fact that in characteristic p we have the Freshman’s dream (x + y)p = xp + y p . 697
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The homomorphism is clearly injective, but is surjective if and only if k is infinite. The failure of surjectivity when k is finite is due to the existence of non-zero polynomials which induce the zero function on k (e.g. xq − x over the finite field with q elements). Even though this ring is not commutative, it still possesses (left and right) division algorithms.
236.3 References • Goss, D. (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 35, Berlin, New York: Springer-Verlag, ISBN 978-3-540-61087-8, MR 1423131, Zbl 0874.11004 • Rosen, Michael (2002), Number Theory in Function Fields, Graduate Texts in Mathematics 210, SpringerVerlag, ISBN 0-387-95335-3, ISSN 0072-5285, Zbl 1043.11079
Chapter 237
u-invariant In mathematics, the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field. The universal invariant u(F) of a field F is the largest dimension of an anisotropic quadratic space over F, or ∞ if this does not exist. Since formally real fields have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields. An equivalent formulation is that u is the smallest number such that every form of dimension greater than u is isotropic, or that every form of dimension at least u is universal.
237.1 Examples • For the complex numbers, u(C) = 1. • If F is quadratically closed then u(F) = 1. • The function field of an algebraic curve over an algebraically closed field has u ≤ 2; this follows from Tsen’s theorem that such a field is quasi-algebraically closed.[1] • If F is a nonreal global or local field, or more generally a linked field, then u(F) = 1,2,4 or 8.[2]
237.2 Properties • If F is not formally real then u(F) is at most q(F ) = F ⋆ /F ⋆2 , the index of the squares in the multiplicative group of F.[3] • u(F) cannot take the values 3, 5, or 7.[4] Fields exist with u = 6[5][6] and u = 9.[7] • Merkurjev has shown that every even integer occurs as the value of u(F) for some F.[8] [9] • The u-invariant is bounded under finite-degree field extensions. If E/F is a field extension of degree n then u(E) ≤
n+1 u(F ) . 2
In the case of quadratic extensions, the u-invariant is bounded by
u(F ) − 2 ≤ u(E) ≤
3 u(F ) 2
and all values in this range are achieved.[10] 699
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237.3 The general u-invariant Since the u-invariant is of little interest in the case of formally real fields, we define a general u-invariant to be the maximum dimension of an anisotropic form in the torsion subgroup of the Witt ring of F, or ∞ if this does exist.[11] For non-formally real fields, the Witt ring is torsion, so this agrees with the previous definition.[12] For a formally real field, the general u-invariant is either even or ∞.
237.3.1
Properties
• u(F) ≤ 1 if and only if F is a Pythagorean field.[12]
237.4 References [1] Lam (2005) p.376 [2] Lam (2005) p.406 [3] Lam (2005) p. 400 [4] Lam (2005) p. 401 [5] Lam (2005) p.484 [6] Lam, T.Y. (1989). “Fields of u-invariant 6 after A. Merkurjev”. Ring theory 1989. In honor of S. A. Amitsur, Proc. Symp. and Workshop, Jerusalem 1988/89. Israel Math. Conf. Proc. 1. pp. 12–30. Zbl 0683.10018. [7] Izhboldin, Oleg T. (2001). “Fields of u-Invariant 9”. Annals of Mathematics, 2 ser 154 (3): 529–587. Zbl 0998.11015. [8] Lam (2005) p. 402 [9] Elman, Karpenko, Merkurjev (2008) p. 170 [10] Mináč, Ján; Wadsworth, Adrian R. (1995). “The u-invariant for algebraic extensions”. In Rosenberg, Alex. K-theory and algebraic geometry: connections with quadratic forms and division algebras. Summer Research Institute on quadratic forms and division algebras, July 6-24, 1992, University of California, Santa Barbara, CA (USA). Proc. Symp. Pure Math. 58.2. Providence, RI: American Mathematical Society. pp. 333–358. Zbl 0824.11018. [11] Lam (2005) p. 409 [12] Lam (2005) p. 410
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022. • Elman, Richard; Karpenko, Nikita; Merkurjev, Alexander (2008). The algebraic and geometric theory of quadratic forms. American Mathematical Society Colloquium Publications 56. American Mathematical Society, Providence, RI. ISBN 978-0-8218-4329-1.
Chapter 238
Unique factorization domain “Unique factorization” redirects here. For the uniqueness of integer factorization, see fundamental theorem of arithmetic. In mathematics, a unique factorization domain (UFD) is a commutative ring in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki. Unique factorization domains appear in the following chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
238.1 Definition Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R can be written as a product (an empty product if x is a unit) of irreducible elements pᵢ of R and a unit u: x = u p1 p2 ... pn with n ≥ 0 and this representation is unique in the following sense: If q1 ,...,qm are irreducible elements of R and w is a unit such that x = w q1 q2 ... qm with m ≥ 0, then m = n, and there exists a bijective map φ : {1,...,n} → {1,...,m} such that pi is associated to qᵩ₍i₎ for i ∈ {1, ..., n}. The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful: A unique factorization domain is an integral domain R in which every non-zero element can be written as a product of a unit and prime elements of R.
238.2 Examples Most rings familiar from elementary mathematics are UFDs: • All principal ideal domains, hence all Euclidean domains, are UFDs. In particular, the integers (also see fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs. 701
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CHAPTER 238. UNIQUE FACTORIZATION DOMAIN
• If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By iteration, a polynomial ring in any number of variables over any UFD (and in particular over a field) is a UFD. • The Auslander–Buchsbaum theorem states that every regular local ring is a UFD. • The formal power series ring K[[X1 ,...,Xn]] over a field K (or more generally over a regular UFD such as a PID) is a UFD. On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if R is the localization of k[x,y,z]/(x2 + y3 + z7 ) at the prime ideal (x,y,z) then R is a local ring that is a UFD, but the formal power series ring R[[X]] over R is not a UFD. • Mori showed that if the completion of a Zariski ring, such as a Noetherian local ring, is a UFD, then the ring is a UFD.[1] The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. The question of when this happens is rather subtle: for example, for the localization of k[x,y,z]/(x2 + y3 + z5 ) at the prime ideal (x,y,z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x,y,z]/(x2 + y3 + z7 ) at the prime ideal (x,y,z) the local ring is a UFD but its completion is not. • Let R be any field of characteristic not 2. Klein and Nagata showed that the ring R[X1 ,...,Xn]/Q is a UFD whenever Q is a nonsingular quadratic form in the X's and n is at least 5. When n=4 the ring need not be a UFD. For example, R[X, Y, Z, W ]/(XY − ZW ) is not a UFD, because the element XY equals the element ZW so that XY and ZW are two different factorizations of the same element into irreducibles. • The ring of formal power series over the complex numbers is factorial, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, e.g.: ) ∞ ( ∏ z2 sin πz = πz 1− 2 . n n=1 • The ring Q[x,y]/(x2 + 2y2 + 1) is factorial, but the ring Q(i)[x,y]/(x2 + 2y2 + 1) is not. On the other hand, The ring Q[x,y]/(x2 + y2 – 1) is not factorial, but the ring Q(i)[x,y]/(x2 + y2 – 1) is (Samuel 1964, p.35). Similarly the coordinate ring R[X,Y,Z]/(X2 + Y 2 + Z 2 − 1) of the 2-dimensional real sphere is factorial, but the coordinate ring C[X,Y,Z]/(X2 + Y 2 + Z 2 − 1) of the complex sphere is not. • Suppose that the variables Xi are given weights wi, and F(X1 ,...,Xn) is a homogeneous polynomial of weight w. Then if c is coprime to w and R is a UFD and either every finitely generated projective module over R is free or c is 1 mod w, the ring R[X1 ,...,Xn,Z]/(Z c − F(X1 ,...,Xn)) is a factorial ring (Samuel 1964, p.31). Non-example: √ √ • The quadratic integer ring Z[ −5] of all complex numbers b −5 ) ( a +√ ) , where a and b are integers, ( of the √ form is not a UFD because 6 factors as both (2)(3) and as 1 + −5 1 − −5 . These truly are different √ √ factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, 1 + −5 , and 1 − −5 are associate. It is not hard to show that all four factors are irreducible as well, though this may not be obvious.[2] See also algebraic integer.
238.3 Properties Some concepts defined for integers can be generalized to UFDs: • In UFDs, every irreducible element is prime. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element z ∈ K[x, y, z]/(z 2 − xy) is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the ACCP is a UFD if and only if every irreducible element is prime.
238.4. EQUIVALENT CONDITIONS FOR A RING TO BE A UFD
703
• Any two (or finitely many) elements of a UFD have a greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d which divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated. • Any UFD is integrally closed. In other words, if R is a UFD with quotient field K, and if an element k in K is a root of a monic polynomial with coefficients in R, then k is an element of R. • Let S be a multiplicatively closed subset of a UFD A. Then the localization S −1 A is a UFD. A partial converse to this also holds; see below.
238.4 Equivalent conditions for a ring to be a UFD A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given below). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case it is in fact a principal ideal domain. There are also equivalent conditions for non-noetherian integral domains. Let A be an integral domain. Then the following are equivalent. 1. A is a UFD. 2. Every nonzero prime ideal of A contains a prime element. (Kaplansky) 3. A satisfies ascending chain condition on principal ideals (ACCP), and the localization S −1 A is a UFD, where S is a multiplicatively closed subset of A generated by prime elements. (Nagata criterion) 4. A satisfies (ACCP) and every irreducible is prime. 5. A is atomic and every irreducible is prime. 6. A is a GCD domain (i.e., any two elements have a greatest common divisor) satisfying (ACCP). 7. A is a Schreier domain,[3] and atomic. 8. A is a pre-Schreier domain and atomic. 9. A has a divisor theory in which every divisor is principal. 10. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) 11. A is a Krull domain and every prime ideal of height 1 is principal.[4] In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since, in a PID, every prime ideal is generated by a prime element. For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains height one prime ideal (induction on height), which is principal. By (2), the ring is a UFD.
238.5 See also • Parafactorial local ring
238.6 References [1] Bourbaki, 7.3, no 6, Proposition 4. [2] Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0.
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[3] A Schreier domain is an integrally closed integral domain where, whenever x divides yz, x can be written as x = x1 x2 so that x1 divides y and x2 divides z. In particular, a GCD domain is a Schreier domain [4] Bourbaki, 7.3, no 2, Theorem 1.
• N. Bourbaki. Commutative algebra. • B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5. Chap. 4. • Chapter II.5 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001 • David Sharpe (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6. • Samuel, Pierre (1964), Murthy, M. Pavman, ed., Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics 30, Bombay: Tata Institute of Fundamental Research, MR 0214579 • Samuel, Pierre (1968). “Unique factorization”. The American Mathematical Monthly 75: 945–952. doi:10.2307/2315529. ISSN 0002-9890.
Chapter 239
Unit (ring theory) Not to be confused with Unit ring. In mathematics, an invertible element or a unit in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.e. an element v such that uv = vu = 1R, where 1R is the multiplicative identity.[1][2] The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the zero ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring. The term unit is also used to refer to the identity element 1R of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. For this reason, some authors call 1R “unity” or “identity”, and say that R is a “ring with unity” or a “ring with identity” rather than a “ring with a unit”. The multiplicative identity 1R and its opposite −1R are always units. Hence, pairs of additive inverse elements[3] x and −x are always associated.
239.1 Group of units Main article: Green’s relations § The H and D relations The units of R form a group U(R) under multiplication, the group of units of R. Other common notations for U(R) are R∗ , R× , and E(R) (for the German term Einheit). In a commutative unital ring R, the group of units U(R) acts on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ∼ on R called associatedness such that r∼s means that there is a unit u with r = us. One can check that U is a functor from the category of rings to the category of groups: every ring homomorphism f : R → S induces a group homomorphism U(f) : U(R) → U(S), since f maps units to units. This functor has a left adjoint which is the integral group ring construction. In an integral domain the cardinality of an equivalence class of associates is the same as that of U(R). A ring R is a division ring if and only if U(R) = R ∖ {0} . 705
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239.2 Examples • In the ring of integers Z, the only units are +1 and −1. • In the ring Z/nZ of integers modulo n, the units are the congruence classes (mod n) represented by integers coprime to n. They constitute the multiplicative group of integers modulo n. • Any root of unity in a ring R is a unit. (If rn = 1, then rn − 1 is a multiplicative inverse of r.) • If R is the ring of integers in a number field, Dirichlet’s unit theorem implies that the unit group of R is a finitely generated abelian group. For example, we have (√5 + 2)(√5 − 2) = 1 in the ring Z[1 + √5/2], and in fact the unit group of this ring is infinite. In general, the unit group of (the ring of integers of) a real quadratic field is infinite (of rank 1). • The unit group of the ring Mn(F) of n × n matrices over a field F is the group GLn(F) of invertible matrices.
239.3 References [1] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. [2] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. [3] In a ring, the additive inverse of a non-zero element can equal to the element itself.
Chapter 240
Universal quadratic form In mathematics, a universal quadratic form is a quadratic form over a ring which represents every element of the ring.[1] A non-singular form over a field which represents zero non-trivially is universal.[2]
240.1 Examples • Over the real numbers, the form x2 in one variable is not universal, as it cannot represent negative numbers: the two-variable form x2 - y2 is universal for R. • Lagrange’s four-square theorem states that every positive integer is the sum of four squares. Hence the form x2 + y2 + z2 + t 2 - u2 is universal for Z. • Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.[3]
240.2 Forms over the rational numbers The Hasse–Minkowski theorem implies that a form is universal over Q if and only if it is universal over Qp for all p (where we include p=∞, letting Q∞ denote R).[4] A form over R is universal if and only if it is not definite; a form over Qp is universal if it has dimension at least 4.[5] We conclude that all indefinite forms of dimension at least 4 over Q are universal.[4]
240.3 See also • The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers.
240.4 References [1] Lam (2005) p.10 [2] Rajwade (1993) p.146 [3] Lam (2005) p.36 [4] Serre (1973) p.43 [5] Serre (1973) p.37
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. 707
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CHAPTER 240. UNIVERSAL QUADRATIC FORM
• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022. • Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics 7. Springer-Verlag. ISBN 0-387-90040-3. Zbl 0256.12001.
Chapter 241
Valuation (algebra) In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. They generalize to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.
241.1 Definition To define the algebraic concept of valuation, the following objects are needed: • a field K and its multiplicative subgroup K × , • an abelian totally ordered group (Γ, +, ≥) (which could also be given in multiplicative notation as (Γ, ·, ≥)). The ordering and group law on Γ are extended to the set Γ ∪ {∞} [1] by the rules • ∞ ≥ α for all α in Γ, • ∞ + α = α + ∞ = ∞ for all α in Γ. Then a valuation of K is any map v : K → Γ ∪ {∞} which satisfies the following properties for all a, b in K: • v(a) = ∞ if, and only if, a = 0, • v(ab) = v(a) + v(b), • v(a + b) ≥ min(v(a), v(b)), with equality if v(a)≠v(b). Some authors use the term exponential valuation rather than “valuation”. In this case the term “valuation” means "absolute value". A valuation v is called trivial (or the trivial valuation of K) if v(a) = 0 for all a in K × , otherwise it is called nontrivial. For valuations used in geometric applications, the first property implies that any non-empty germ of an analytic variety near a point contains that point. The second property asserts that any valuation is a group homomorphism, while the third property is a translation of the triangle inequality from metric spaces to ordered groups. It is possible to give a dual definition of the same concept using the multiplicative notation for Γ: if, instead of ∞, an element O[2] is given and the ordering and group law on Γ are extended by the rules 709
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CHAPTER 241. VALUATION (ALGEBRA)
• O ≤ α for all α in Γ, • O · α = α · O = O for all α in Γ, then a valuation of K is any map v : K → Γ ∪ {O} satisfying the following properties for all a, b in K: • v(a) = O if, and only if, a = 0, • v(ab) = v(a) · v(b), • v(a + b) ≤ max(v(a), v(b)), with equality if v(a)≠v(b). (Note that in this definition, the directions of the inequalities are reversed.) A valuation is commonly assumed to be surjective, since many arguments used in ordinary mathematical research involving those objects use preimages of unspecified elements of the ordered group contained in its codomain. Also, the first definition of valuation given is more frequently encountered in ordinary mathematical research, thus it is the only one used in the following considerations and examples.
241.1.1
Associated objects
If v : K → Γ ∪ {∞} is a valuation, then there are several objects that can be defined from it: • the value group of v (or valuation group of v), denoted Γv, is v(K × ), it is a subgroup of Γ, • the valuation ring of v, denoted Rv is the set of elements a of K such that v(a) ≥ 0, it is a valuation ring, • the prime ideal of v (or the maximal ideal of v), denoted mv is the set of elements a of K such that v(a) > 0, it is a maximal ideal of Rv, • the residue field of v, denoted kv is Rv/mv, it is a field. • a place of K into kv ∪ {∞}, whose restriction to Rv is the natural projection.
241.2 Basic properties 241.2.1
Equivalence of valuations
Two valuations v1 and v2 of K with valuation group Γ1 and Γ2 , respectively, are said to be equivalent if there is an order-preserving group isomorphism φ : Γ1 → Γ2 such that v2 (a) = φ(v1 (a)) for all a in K × . This is an equivalence relation. Two valuations of K are equivalent if, and only if, they have the same valuation ring. An equivalence class of valuations of a field is called a place. Ostrowski’s theorem gives a complete classification of places of the field of rational numbers Q: these are precisely the equivalence classes of valuations for the p-adic completions of Q.
241.2.2
Extension of valuations
Let v be a valuation of K and let L be a field extension of K. An extension of v (to L) is a valuation w of L such that the restriction of w to K is v. The set of all such extensions is studied in the ramification theory of valuations. Let L/K be a finite extension and let w be an extension of v to L. The index of Γv in Γw, e(w/v) = [Γw : Γv], is called the reduced ramification index of w over v. It satisfies e(w/v) ≤ [L : K] (the degree of the extension L/K). The relative degree of w over v is defined to be f(w/v) = [Rw/mw : Rv/mv] (the degree of the extension of residue fields). It is also less than or equal to the degree of L/K. When L/K is separable, the ramification index of w over v is defined to be e(w/v)pi , where pi is the inseparable degree of the extension Rw/mw over Rv/mv.
241.3. EXAMPLES
241.2.3
711
Complete valued fields
When the ordered abelian group Γ is the additive group of the integers, the associated valuation induces a metric on the field K. If K is complete with respect to this metric, then it is called a complete valued field. In general, a valuation induces a uniform structure on K, and K is called a complete valued field if it is complete as a uniform space. There is a related property known as spherical completeness: it is equivalent to completeness if Γ = Z, but stronger in general.
241.3 Examples 241.3.1
π-adic valuation
Let R be a principal ideal domain, K be its field of fractions, and π be an irreducible element of R. Since every principal ideal domain is a unique factorization domain, every non-zero element a of R can be written (essentially) uniquely as
a = π ea pe11 pe22 · · · penn where the e's are non-negative integers and the pi are irreducible elements of R that are not associates of π. In particular, the integer ea is uniquely determined by a. The π-adic valuation of K is then given by • vπ (0) = ∞ • vπ (a/b) = ea − eb , for a, b ∈ R, a, b ̸= 0. If π' is another irreducible element of R such that (π') = (π) (that is, they generate the same ideal in R), then the π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called the P-adic valuation, where P = (π). When R = Z, then K = Q, and π is some prime number p (or its negative). The π-adic valuation obtained is the p-adic valuation on Q.
241.3.2 P-adic valuation on a Dedekind domain The previous example can be generalized to Dedekind domains. Let R be a Dedekind domain, K its field of fractions, and let P be a non-zero prime ideal of R. Then, the localization of R at P, denoted RP, is a principal ideal domain whose field of fractions is K. The construction of the previous section applied to the prime ideal PRP of RP yields the P-adic valuation of K.
241.3.3
Geometric notion of contact
Let C[x, y], C(x, y) be the ring of complex polynomials of two variables and the field of complex rational functions respectively. Consider the (convergent) power series
f (x, y) = y −
∞ ∑
xn ∈ C{x, y}
n=3
whose zero set, the analytic variety V , can be parametrized by one coordinate t as follows
{ } Vf = (x, y) ∈ C2 : f (x, y) = 0 =
{
(
(x, y) ∈ C : (x, y) = 2
t,
∞ ∑ n=3
)} t
n
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CHAPTER 241. VALUATION (ALGEBRA)
It is possible to define a map v : C[x, y] → Z as the value of the order of the formal power series in the variable t obtained by restriction of any polynomial P in C[x, y] to the points of the set V
( ) v(P ) = ordt P |Vf = ordt
( ( P
t,
∞ ∑
)) n
t
,
∀P ∈ C[x, y]
n=3
It is also possible to extend the map v from its original ring of definition to the whole field C(x, y) as follows { v(P ) − v(Q) v(P /Q) = ∞
∗
P /Q ∈ C(x, y) P ≡ 0 ∈ C(x, y)
As the power series f is not a polynomial, it is easy to prove that the extended map v is a valuation: the value v(P) is called intersection number between the curves (1-dimensional analytic varieties) VP and V . As an example, the computation of some intersection numbers follows:
v(x) = ordt (t) = 1 ( ) ( ) v(x − y 2 ) = ordt t6 − t6 − 2t7 − 3t8 − · · · = ordt −2t7 − 3t8 − · · · = 7 ( 6 2) ( ) = ordt −2t7 − 3t8 − · · · − ordt (t) v x −y x 6
=7−1=6
241.4 Vector spaces over valuation fields Suppose that Γ is the set of non-negative real numbers. Then we say that the valuation is non-discrete if its range is not finite. Suppose that X is a vector space over K and that A and B are subsets of X. Then we say that A absorbs B if there exists a α in K such that λ in K and |λ| ≥ |α| implies that B λ A. A is called radial or absorbing if A absorbs every finite subset of X. Radial subsets of X are invariant under finite intersection. And A is called circled if λ in K and |λ| ≥ |α| implies λ A A. The set of circled subsets of L is invariant under arbitrary intersections. The circled hull of A is the intersection of all circled subsets of X containing A. Suppose that X and Y are vector spaces over a non-discrete valuation field K, let A X, B Y, and let f : X → Y be a linear map. If B is circled or radial then so is f −1 (B) . If A is circled then so is f(A) but if A is radial then f(A) will be radial under the additional condition that f is surjective.
241.5 See also • Discrete valuation • Euclidean valuation • Valuation (measure theory) • Valuation algebra
241.6 Notes [1] The symbol ∞ denotes an element not in Γ, and has not any other meaning. Its properties are simply defined by axioms, as in every formal presentation of a mathematical theory. [2] As for the symbol ∞, O denotes an element not in Γ and has not any other meaning, its properties being again defined by axioms.
241.7. REFERENCES
713
241.7 References • Efrat, Ido (2006), Valuations, orderings, and Milnor K-theory, Mathematical Surveys and Monographs 124, Providence, RI: American Mathematical Society, ISBN 0-8218-4041-X, Zbl 1103.12002 • Jacobson, Nathan (1989) [1980], “Valuations: paragraph 6 of chapter 9”, Basic algebra II (2nd ed.), New York: W. H. Freeman and Company, ISBN 0-7167-1933-9, Zbl 0694.16001. A masterpiece on algebra written by one of the leading contributors. • Chapter VI of Zariski, Oscar; Samuel, Pierre (1976) [1960], Commutative algebra, Volume II, Graduate Texts in Mathematics 29, New York, Heidelberg: Springer-Verlag, ISBN 978-0-387-90171-8, Zbl 0322.13001 • Schaefer, Helmuth H.; Wolff, M.P. (1999). Topological Vector Spaces. GTM 3. New York: Springer-Verlag. pp. 10–11. ISBN 9780387987262.
241.8 External links • Danilov, V.I. (2001), “Valuation”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 9781-55608-010-4 • Discrete valuation at PlanetMath.org. • Valuation at PlanetMath.org. • Weisstein, Eric W., “Valuation”, MathWorld.
Chapter 242
Valuation ring In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D. Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring. The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially ordered by dominance or refinement,[1] where (A, mA ) dominates (B, mB ) if A ⊃ B and mA ∩ B = mB .[2] Every local ring in a field K is dominated by some valuation ring of K. An integral domain whose localization at any prime ideal is a valuation ring is called a Prüfer domain.
242.1 Examples • Any field is a valuation ring. • Z₍p₎, the localization of the integers Z at the prime ideal (p), consisting of ratios where the numerator is any integer and the denominator is not divisible by p. The field of fractions is the field of rational numbers Q. • The ring of meromorphic functions on the entire complex plane which have a Maclaurin series (Taylor series expansion at zero) is a valuation ring. The field of fractions are the functions meromorphic on the whole plane. If f does not have a Maclaurin series then 1/f does. • Any ring of p-adic integers Zp for a given prime p is a local ring, with field of fractions the p-adic numbers Q . The integral closure Zpcl of the p-adic integers is also a local ring, with field of fractions Qpcl (the algebraic closure of p-adic numbers). Both Zp and Zpcl are valuation rings. • Let k be an ordered field. An element of k is called finite if it lies between two integers n
242.2. DEFINITIONS
715
242.2 Definitions There are several equivalent definitions of valuation ring (see below for the characterization in terms of dominance). For a subring D of its field of fractions K the following are equivalent: 1. For every nonzero x in K, either x in D or x−1 in D. 2. The ideals of D are totally ordered by inclusion. 3. The principal ideals of D are totally ordered by inclusion (i.e., the elements in D are totally ordered by divisibility.) 4. There is a totally ordered abelian group Γ (called the value group) and a surjective group homomorphism (called the valuation) ν:K × → Γ with D = { x in K × : ν(x) ≥ 0 } ∪ {0} The equivalence of the first three definitions follows easily. A theorem of (Krull 1939) states that any ring satisfying the first three conditions satisfies the fourth: take Γ to be the quotient K × /D× of the unit group of K by the unit group of D, and take ν to be the natural projection. We can turn Γ into a totally ordered group by declaring the residue classes of elements of D as “positive”.[3] Even further, given any totally ordered abelian group Γ, there is a valuation ring D with value group Γ (see a section below). From the fact that the ideals of a valuation ring are totally ordered, one can conclude that a valuation ring is a local domain, and that every finitely generated ideal of a valuation ring is principal (i.e., a valuation ring is a Bézout domain). In fact, it is a theorem of Krull that an integral domain is a valuation ring if and only if it is a local Bézout domain.[4] It also follows from this that a valuation ring is Noetherian if and only if it is a principal ideal domain. In this case, it is either a field or it has exactly one non-zero maximal ideal; such a valuation ring is called a discrete valuation ring. (By convention, a field is not a discrete valuation ring.) A value group is called discrete if it is isomorphic to the additive group of the integers, and a valuation ring has a discrete valuation group if and only if it is a discrete valuation ring.[5] Very rarely, valuation ring may refer to a ring that satisfies the second or third condition but is not necessarily a domain. A more common term for this type of ring is "uniserial ring".
242.3 Construction For a given totally ordered abelian group Γ and a residue field k, define K = k((Γ)) to be the ring of formal power series whose powers come from Γ, that is, the elements of K are functions from Γ to k such that the support (the elements of Γ where the function value is not the zero of k) of each function is a well-ordered subset of G. Addition is pointwise, and multiplication is the Cauchy product or convolution, that is the natural operation when viewing the functions as power series: ∑ g∈G
f (g)xg with xg · xh = xg+h .
The valuation ν(f) for f in K is defined to be the least element of the support of f, that is the least element g of Γ such that f(g) is nonzero. The f with ν(f)≥0 (along with 0 in K), form a subring D of K that is a valuation ring with value group Γ, valuation ν, and residue field k. This construction is detailed in (Fuchs & Salce 2001, pp. 66–67), and follows a construction of (Krull 1939) which uses quotients of polynomials instead of power series.
242.4 Dominance and integral closure The units, or invertible elements, of a valuation ring are the elements x such that x −1 is also a member of D. The other elements of D, called nonunits, do not have an inverse, and they form an ideal M. This ideal is maximal among the (totally ordered) ideals of D. Since M is a maximal ideal, the quotient ring D/M is a field, called the residue field of D.
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CHAPTER 242. VALUATION RING
In general, we say a local ring (S, mS ) dominates a local ring (R, mR ) if S ⊃ R and mS ∩ R = mR ; in other words, the inclusion R ⊂ S is a local ring homomorphism. Every local ring (A, p) in a field K is dominated by some valuation ring of K. Indeed, the set consisting of all subrings R of K containing A and 1 ̸∈ pR is nonempty and is inductive; thus, has a maximal element R by Zorn’s lemma. We claim R is a valuation ring. R is a local ring with maximal ideal containing pR by maximality. Again by maximality it is also integrally closed. Now, if x ̸∈ R , then, by maximality, pR[x] = R[x] and thus we can write:
1 = r0 + r1 x + · · · + rn xn ,
ri ∈ pR
Since 1 − r0 is a unit element, this implies that x−1 is integral over R; thus is in R. This proves R is a valuation ring. (R dominates A since its maximal ideal contains p by construction.) A local ring R in a field K is a valuation ring if and only if it is a maximal element of the set of all local rings contained in K partially ordered by dominance. This easily follows from the above.[6] Let A be a subring of a field K and f : A → k a ring homomorphism into an algebraically closed field k. Then f extends to a ring homomorphism g : D → k , D some valuation ring of K containing A. (Proof: Let g : R → k be a maximal extension, which clearly exists by Zorn’s lemma. By maximality, R is a local ring with maximal ideal containing the kernel of f. If S is a local ring dominating R, then S is algebraic over R; if not, S contains a polynomial ring R[x] to which g extends, a contradiction to maximality. It follows S/mS is an algebraic field extension of R/mR . Thus, S → S/mS ,→ k extends g; hence, S = R.) If a subring R of a field K contains a valuation ring D of K, then, by checking Definition 1, R is also a valuation ring of K. In particular, R is local and its maximal ideal contracts to some prime ideal of D, say, p . Then R = Dp since R dominates Dp , which is a valuation ring since the ideals are totally ordered. This observation is subsumed to the following:[7] there is a bijective correspondence p 7→ Dp , Spec(D) → the set of all subrings of K containing D. In particular, D is integrally closed,[8][9] and the Krull dimension of D is the cardinality of proper subrings of K containing D. In fact, the integral closure of an integral domain A in the field of fractions K of A is the intersection of all valuation rings of K containing A.[10] Indeed, the integral closure is contained in the intersection since the valuation rings are integrally closed. Conversely, let x be in K but not integral over A. Since the ideal x−1 A[x−1 ] is not A[x−1 ] ,[11] it is contained in a maximal ideal p . Then there is a valuation ring R that dominates the localization of A[x−1 ] at p . Since x−1 ∈ mR , x ̸∈ R . The dominance is used in algebraic geometry. Let X be an algebraic variety over a field k. Then we say a valuation ring R in k(X) has “center x on X" if R dominates the local ring Ox,X of the structure sheaf at x.[12]
242.5 Ideals in valuation rings We may describe the ideals in the valuation ring by means of its value group. Let Γ be a totally ordered abelian group. A subset Δ of Γ is called a segment if it is nonempty and, for any α in Δ, any element between -α and α is also in Δ (end points included). A subgroup of Γ is called an isolated subgroup if it is a segment and is a proper subgroup. Let D be a valuation ring with valuation v and value group Γ. For any subset A of D, we let ΓA be the complement of the union of v(A − 0) and −v(A − 0) in Γ . If I is a proper ideal, then ΓI is a segment of Γ . In fact, the mapping I 7→ ΓI defines an inclusion-reversing bijection between the set of proper ideals of D and the set of segments of Γ .[13] Under this correspondence, the nonzero prime ideals of D correspond bijectively to the isolated subgroups of Γ. Example: The ring of p-adic integers Zp is a valuation ring with value group Z . The zero subgroup of Z corresponds to the unique maximal ideal (p) ⊂ Zp and the whole group to the zero ideal. The maximal ideal is the only isolated subgroup of Z . The set of isolated subgroups is totally ordered by inclusion. The height or rank r(Γ) of Γ is defined to be the cardinality of the set of isolated subgroups of Γ. Since the nonzero prime ideals are totally ordered and they correspond to isolated subgroups of Γ, the height of Γ is equal to the Krull dimension of the valuation ring D associated with Γ. The most important special case is height one, which is equivalent to Γ being a subgroup of the real numbers ℝ under addition (or equivalently, of the positive real numbers ℝ+ under multiplication.) A valuation ring with a valuation of
242.6. PLACES
717
height one has a corresponding absolute value defining an ultrametric place. A special case of this are the discrete valuation rings mentioned earlier. The rational rank rr(Γ) is defined as the rank of the value group as an abelian group
dimQ (Γ ⊗Z Q)
242.6 Places The reference to this section is Zariski–Samuel. A place of a field K is a ring homomorphism p from a valuation ring D of K to some field such that, for any x ̸∈ D , p(1/x) = 0 . The image of a place is a field called the residue field of p. For example, the canonical map D → D/mD is a place. Example: Let A be a Dedekind domain and p a prime ideal. Then the canonical map Ap → k(p) is a place. We say a place p specializes to a place p', denoted by p ⇝ p′ , if the valuation ring of p contains the valuation ring of p'. In algebraic geometry, we say a prime ideal p specializes to p′ if p ⊂ p′ . The two notions coincide: p ⇝ p′ if and only if a prime ideal corresponding to p specializes to a prime ideal corresponding to p' in some valuation ring (recall that if D ⊃ D′ are valuation rings of the same field, then D corresponds to a prime ideal of D′ .) It can be shown: if p ⇝ p′ , then p′ = q ◦ p|D′ for some place q of the residue field k(p) of p. (Observe p(D′ ) is a valuation ring of k(p) and let q be the corresponding place; the rest is mechanical.) If D is a valuation ring of p, then its Krull dimension is the cardinarity of the specializations other than p to p. Thus, for any place p with valuation ring D of a field K over a field k, we have:
tr. degk k(p) + dim D ≤ tr. degk K If p is a place and A is a subring of the valuation ring of p, then ker(p) ∩ A is called the center of p in A.
242.7 Notes [1] Hartshone 1977, Theorem I.6.1A [2] Efrat (2006) p.55 [3] More precisely, Γ is totally ordered by defining [x] ≥ [y] if and only if xy −1 ∈ D where [x] and [y] are equivalence classes in Γ. cf. Efrat (2006) p.39 [4] Cohn 1968, Proposition 1.5 [5] Efrat (2006) p.43 [6] Proof: if R is a maximal element, then it is dominated by a valuation ring; thus, it itself must be a valuation ring. Conversely, let R be a valuation ring and S a local ring that dominates R but not R. There is x that is in S but not in R. Then x−1 is in R and in fact in the maximal ideal of R. But then x−1 ∈ mS , which is absurd. Hence, there cannot be such S. [7] Zariski−Samuel, Ch. VI, Theorem 3 [8] Efrat (2006) p.38 [9] To see more directly that valuation rings are integrally closed, suppose that xn + a1 xn − 1 + ... + a0 = 0. Then dividing by xn−1 gives us x = − a1 − ... − a0 x − n + 1 . If x were not in D, then x −1 would be in D and this would express x as a finite sum of elements in D, so that x would be in D, a contradiction. [10] Matsumura 1986, Theorem 10.4 [11] In general, x−1 is integral over A if and only if xA[x] = A[x]. [12] Hartshorne 1977, Ch II. Exercise 4.5 [13] Zariski−Samuel, Ch. VI, Theorem 15
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242.8 References • Nicolas Bourbaki, Commutative Algebra, Addison-Wesley, 1972 • Cohn, P. M. (1968), “Bezout rings and their subrings” (PDF), Proc. Cambridge Philos. Soc. 64: 251–264, doi:10.1017/s0305004100042791, ISSN 0008-1981, MR 0222065 (36 #5117), Zbl 0157.08401 • Efrat, Ido (2006), Valuations, orderings, and Milnor K-theory, Mathematical Surveys and Monographs 124, Providence, RI: American Mathematical Society, ISBN 0-8218-4041-X, Zbl 1103.12002 • Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715, Zbl 0973.13001 • Krull, Wolfgang (1939), “Beiträge zur Arithmetik kommutativer Integritätsbereiche. VI. Der allgemeine Diskriminantensatz. Unverzweigte Ringerweiterungen”, Mathematische Zeitschrift 45 (1): 1–19, doi:10.1007/BF01580269, ISSN 0025-5874, MR 1545800, Zbl 0020.34003 • Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: SpringerVerlag, ISBN 978-0-387-90244-9, MR 0463157 • Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Translated from the Japanese by Miles Reid (Second edition ed.), ISBN 0-521-36764-6, Zbl 0666.13002 • Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876
Chapter 243
Weil group For other uses, see Weil–Châtelet group, Mordell–Weil group, and Weyl group. In mathematics, a Weil group, introduced by Weil (1951), is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is generally denoted WF. There also exists “finite level” modifications of the Galois groups: if E/F is a finite extension, then the relative Weil group of E/F is WE/F = WF/W c E (where the superscript c denotes the commutator subgroup). For more details about Weil groups see (Artin & Tate 2009) or (Tate 1979) or (Weil 1951).
243.1 Weil group of a class formation The Weil group of a class formation with fundamental classes uE/F ∈ H 2 (E/F, AF ) is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program. If E/F is a normal layer, then the (relative) Weil group WE/F of E/F is the extension 1 → AF → WE/F → Gal(E/F) → 1 corresponding (using the interpretation of elements in the second group cohomology as central extensions) to the fundamental class uE/F in H 2 (Gal(E/F), AF ). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers G/F, for F an open subgroup of G. The reciprocity map of the class formation (G, A) induces an isomorphism from AG to the abelianization of the Weil group.
243.2 Weil group of an archimedean local field For archimedean local fields the Weil group is easy to describe: for C it is the group C× of non-zero complex numbers, and for R it is a non-split extension of the Galois group of order 2 by the group of non-zero complex numbers, and can be identified with the subgroup C× ∪ j C× of the non-zero quaternions.
243.3 Weil group of a finite field For finite fields the Weil group is infinite cyclic. A distinguished generator is provided by the Frobenius automorphism. Certain conventions on terminology, such as arithmetic Frobenius, trace back to the fixing here of a generator (as the Frobenius or its inverse). 719
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243.4 Weil group of a local field For a local field of characteristic p > 0, the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields). For p-adic fields the Weil group is a dense subgroup of the absolute Galois group, and consists of all elements whose image in the Galois group of the residue field is an integral power of the Frobenius automorphism. More specifically, in these cases, the Weil group does not have the subspace topology, but rather a finer topology. This topology is defined by giving the inertia subgroup its subspace topology and imposing that it be an open subgroup of the Weil group. (The resulting topology is "locally profinite".)
243.5 Weil group of a function field For global fields of characteristic p>0 (function fields), the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).
243.6 Weil group of a number field For number fields there is no known “natural” construction of the Weil group without using cocycles to construct the extension. The map from the Weil group to the Galois group is surjective, and its kernel is the connected component of the identity of the Weil group, which is quite complicated.
243.7 Weil–Deligne group The Weil–Deligne group scheme (or simply Weil–Deligne group) W′K of a non-archimedean local field, K, is an extension of the Weil group WK by a one-dimensional additive group scheme Ga, introduced by Deligne (1973, 8.3.6). In this extension the Weil group acts on the additive group by
wxw−1 = ||w||x ||w||
where w acts on the residue field of order q as a→aq . The local Langlands correspondence for GLn over K (now proved) states that there is a natural bijection between isomorphism classes of irreducible admissible representations of GLn(K) and certain n-dimensional representations of the Weil–Deligne group of K. The Weil–Deligne group often shows up through its representations. In such cases, the Weil–Deligne group is sometimes taken to be WK × SL(2,C) or WK × SU(2,R), or is simply done away with and Weil–Deligne representations of WK are used instead.[1] In the archimedean case, the Weil–Deligne group is simply defined to be Weil group.
243.8 See also • Langlands group • Shafarevich–Weil theorem
243.9 Notes [1] Rohrlich 1994
243.10. REFERENCES
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243.10 References • Artin, Emil; Tate, John (2009) [1952], Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, MR 0223335 • Deligne, Pierre (1973), “Les constantes des équations fonctionnelles des fonctions L”, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture notes in mathematics 349, Berlin, New York: Springer-Verlag, pp. 501–597, doi:10.1007/978-3-540-37855-6_7, MR 0349635 • Kottwitz, Robert (1984), “Stable trace formula: cuspidal tempered terms”, Duke Mathematical Journal 51 (3): 611–650, doi:10.1215/S0012-7094-84-05129-9, MR 0757954 • Rohrlich, David (1994), “Elliptic curves and the Weil–Deligne group”, in Kisilevsky, Hershey; Murty, M. Ram, Elliptic curves and related topics, CRM Proceedings and Lecture Notes 4, American Mathematical Society, ISBN 978-0-8218-6994-9 • Tate, J. (1979), “Number theoretic background”, Automorphic forms, representations, and L-functions Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 0-8218-1435-4 • Weil, André (1951), “Sur la theorie du corps de classes (On class field theory)", Journal of the Mathematical Society of Japan 3: 1–35, doi:10.2969/jmsj/00310001, ISSN 0025-5645, reprinted in volume I of his collected papers, ISBN 0-387-90330-5
Chapter 244
Zahlbericht In mathematics, the Zahlbericht (number report) was a report on algebraic number theory by Hilbert (1897, 1998, (English translation)).
244.1 History Corry (1996) and Schappacher (2005) and the English introduction to (Hilbert 1998) give detailed discussions of the history and influence of Hilbert’s Zahlbericht. Some earlier reports on number theory include the report by H. J. S. Smith in 6 parts between 1859 and 1865, reprinted in Smith (1965), and the report by Brill & Noether (1894). Hasse (1926, 1927, 1930) wrote an update of Hilbert’s Zahlbericht that covered class field theory (republished in 1 volume as (Hasse 1970)). In 1893 the German mathematical society invited Hilbert and Minkowski to write reports on the theory of numbers. They agreed that Minkowski would cover the more elementary parts of number theory while Hilbert would cover algebraic number theory. Minkowski eventually abandoned his report, while Hilbert’s report was published in 1897. It was reprinted in volume 1 of his collected works, and republished in an English translation in 1998.
244.2 Contents Part 1 covers the theory of general number fields, including ideals, discriminants, differents, units, and ideal classes. Part 2 covers Galois number fields, including in particular Hilbert’s theorem 90. Part 3 covers quadratic number fields, including the theory of genera, and class numbers of quadratic fields. Part 4 covers cyclotomic fields, including the Kronecker–Weber theorem (theorem 131), the Hilbert–Speiser theorem (theorem 132), and the Eisenstein reciprocity law for lth power residues (theorem 140) . Part 5 covers Kummer number fields, and ends with Kummer’s proof of Fermat’s last theorem for regular primes.
244.3 References • Brill, A.; Noether, M. (1894), “Die Entwickelung der Theorie der algebraischen Functionen in älterer und neuerer Zeit”, Jahresbericht der Deutschen Mathematiker-Vereinigung (in German) 3: 107–566, ISSN 00120456 • Corry, Leo (1996), Modern algebra and the rise of mathematical structures, Science Networks. Historical Studies 17, Birkhäuser Verlag, ISBN 978-3-7643-5311-7, MR 1391720 • Hasse, H. (1926), “Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie.”, Jahresbericht der Deutschen Mathematiker-Vereinigung (in German) 35: 1–55 722
244.4. EXTERNAL LINKS
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• Hasse, H. (1927), “Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil Ia: Beweise zu I.”, Jahresbericht der Deutschen Mathematiker-Vereinigung (in German) 36: 233–311 • Hasse, H. (1930), “Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz”, Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), Ergänzungsband 6 • Hasse, Helmut (1970) [1930], Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil I: Klassenkörpertheorie. Teil Ia: Beweise zu Teil I. Teil II: Reziprozitätsgesetz (3rd ed.), Physica-Verlag, ISBN 978-3-7908-0010-4, MR 0266893 • Hilbert, David (1897), “Die Theorie der algebraischen Zahlkörper”, Jahresbericht der Deutschen MathematikerVereinigung (in German) 4: 175–546, ISSN 0012-0456 • Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, ISBN 9783-540-62779-1, MR 1646901 • Schappacher, N. (2005), “Chapter 54. David Hilbert, Report on algebraic number fields”, in Grattan-Guinness, Ivor, Landmark writings in western mathematics 1640–1940, Elsevier B. V., Amsterdam, ISBN 978-0-44450871-3, MR 2169816 • Smith, Henry John Stephen (1965) [1894], Glaisher, J. W. L., ed., The Collected Mathematical Papers of Henry John Stephen Smith I, New York: AMS Chelsea Publishing, ISBN 978-0-8284-0187-6, volume 1volume 2
244.4 External links • Introduction to the English Edition of Hilbert’s Zahlbericht
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244.5 Text and image sources, contributors, and licenses 244.5.1
Text
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Davidsevilla, The Return of WWWBot, Yobot, Brad7777, Deltahedron, K9re11 and Anonymous: 1 • Abstract analytic number theory Source: https://en.wikipedia.org/wiki/Abstract_analytic_number_theory?oldid=650248928 Contributors: Michael Hardy, Revolver, Oleg Alexandrov, John Z, SmackBot, Dreadstar, CRGreathouse, Vanish2, JackSchmidt, Jncraton, Omnipedian, Lightbot, Citation bot, Brad7777, Deltahedron, Spectral sequence and Anonymous: 6 • Additive polynomial Source: https://en.wikipedia.org/wiki/Additive_polynomial?oldid=659261624 Contributors: Michael Hardy, Evercat, Charles Matthews, Giftlite, Drbreznjev, Oleg Alexandrov, Linas, R.e.b., Crisco 1492, SmackBot, RDBury, Stootoon, LokiClock, Nusumareta, Quondum, Escspeed, Idonotwanttoknow and Anonymous: 5 • Adele ring Source: https://en.wikipedia.org/wiki/Adele_ring?oldid=671915801 Contributors: SimonP, Stevertigo, Michael Hardy, Schneelocke, Charles Matthews, ComplexZeta, Giftlite, Abiola Lapite, EmilJ, Eric Kvaalen, Joriki, Salix alba, R.e.b., Gwaihir, Crasshopper, SmackBot, Vina-iwbot~enwiki, Tesseran, Aghitza, CRGreathouse, Oo7565, Gogo Dodo, RobHar, Liquid-aim-bot, Deflective, Forgetfulfunctor, PhiTower, Yill577, Jakob.scholbach, LokiClock, Hesam7, Geometry guy, Andreas Carter, CàlculIntegral, Addbot, Roentgenium111, Yobot, AnomieBOT, Xqbot, Omnipaedista, Point-set topologist, Sławomir Biały, Tviilo, Zygmuund, Chrangers, Symmetries134, Deltahedron, Spectral sequence and Anonymous: 20 • Adelic algebraic group Source: https://en.wikipedia.org/wiki/Adelic_algebraic_group?oldid=643060864 Contributors: Edward, Michael Hardy, TakuyaMurata, Charles Matthews, Drbreznjev, Oleg Alexandrov, R.e.b., SmackBot, Bluebot, CRGreathouse, Headbomb, RobHar, Deflective, Ahuskay, JackSchmidt, Alecobbe, Addbot, Roentgenium111, Wohingenau, Omnipaedista, Concord113, Deltahedron and Anonymous: 10 • Adjunction (field theory) Source: https://en.wikipedia.org/wiki/Adjunction_(field_theory)?oldid=634297202 Contributors: Zundark, TakuyaMurata, MathMartin, That Guy, From That Show!, Danpovey, Ntsimp, David Eppstein, JackSchmidt, Addbot, AnomieBOT, OnePt618, Anita5192 and Anonymous: 3 • Albert–Brauer–Hasse–Noether theorem Source: https://en.wikipedia.org/wiki/Albert%E2%80%93Brauer%E2%80%93Hasse%E2% 80%93Noether_theorem?oldid=663673902 Contributors: Michael Hardy, Phil Boswell, Giftlite, Rjwilmsi, R.e.b., JohnCD, Headbomb, Arcfrk, Moonriddengirl, Henry Delforn (old), Ozob, Yobot, Citation bot, Citation bot 1, Jonesey95, Dion Boucicault, Brad7777, Deltahedron, Spectral sequence, Nalbert2 and Anonymous: 1 • Algebraic closure Source: https://en.wikipedia.org/wiki/Algebraic_closure?oldid=653597774 Contributors: Damian Yerrick, AxelBoldt, Mav, Zundark, Enchanter, Michael Hardy, Charles Matthews, Dysprosia, Tobias Bergemann, Giftlite, DefLog~enwiki, Vivacissamamente, Rich Farmbrough, Haham hanuka, HasharBot~enwiki, Oleg Alexandrov, FlaBot, YurikBot, Gslin, SmackBot, Jushi, Bluebot, Gutworth, NeilFraser, MvH, RekishiEJ, Gregbard, Salgueiro~enwiki, JAnDbot, David Eppstein, Aram33~enwiki, DorganBot, Kriega, Ideal gas equation, MystBot, Legobot, Luckas-bot, Yobot, AnomieBOT, Howard McCay, MaximalIdeal, Spaetzle, Deltahedron, SantiLak and Anonymous: 21 • Algebraic extension Source: https://en.wikipedia.org/wiki/Algebraic_extension?oldid=653316776 Contributors: AxelBoldt, Zundark, SimonP, Alodyne, TakuyaMurata, Charles Matthews, Dysprosia, MathMartin, Giftlite, Sim~enwiki, Wmahan, DefLog~enwiki, Vina, Klemen Kocjancic, Rich Farmbrough, Andi5, EmilJ, Mdd, Drbreznjev, Oleg Alexandrov, Banus, SmackBot, Maksim-e~enwiki, Unyoyega, MalafayaBot, Cícero, Vina-iwbot~enwiki, Will Beback, MvH, Jim.belk, Mets501, CmdrObot, CBM, Thijs!bot, RobHar, Vanish2, Trumpet marietta 45750, Linefeed, JackSchmidt, Ideal gas equation, He7d3r, Hans Adler, Legobot, Luckas-bot, Sz-iwbot, Grinevitski, RjwilmsiBot, EmausBot, Wcherowi, ChrisGualtieri, Deltahedron, Bg9989, K9re11 and Anonymous: 14 • Algebraic function field Source: https://en.wikipedia.org/wiki/Algebraic_function_field?oldid=656367650 Contributors: AxelBoldt, Michael Hardy, Charles Matthews, Bearcat, Malcolma, MvH, David Eppstein, AnomieBOT, D.Lazard, Qetuth, Enyokoyama and Anonymous: 1 • Algebraic number field Source: https://en.wikipedia.org/wiki/Algebraic_number_field?oldid=661431528 Contributors: AxelBoldt, Zundark, XJaM, Michael Hardy, TakuyaMurata, Looxix~enwiki, Silverfish, Revolver, Charles Matthews, Dysprosia, Shizhao, Romanm, Rholton, Casito, Tobias Bergemann, Giftlite, Gene Ward Smith, Fropuff, Dratman, Johnflux, Vivacissamamente, Guanabot, Msh210, Oleg Alexandrov, Linas, Marudubshinki, Salix alba, Mathbot, Small potato, YurikBot, Wavelength, Michael Slone, Doetoe, CWenger, Matikkapoika~enwiki, Betacommand, Nbarth, Daqu, Loadmaster, Valoem, Vaughan Pratt, CRGreathouse, Myasuda, Xantharius, JamesAM, Thijs!bot, Headbomb, RobHar, Nick Number, Escarbot, Arch dude, Smartcat, Jakob.scholbach, David Eppstein, Yonidebot, Jacksonwalters, 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Schmidt, DYLAN LENNON~enwiki, SmackBot, Incnis Mrsi, Melchoir, KocjoBot~enwiki, JCSantos, Silly rabbit, Can't sleep, clown will eat me, Dreadstar, Lambiam, Loadmaster, Vaughan Pratt, MC10, Oerjan, Mollwollfumble, Arcfrk, WereSpielChequers, Ideal gas equation, Cliff, WestwoodMatt, Tlepp, Makotoy, Marc van Leeuwen, Addbot, Poco a poco, Luckas-bot, Yobot, Ptbotgourou, AnomieBOT, Rubinbot, Xqbot, GrouchoBot, Raulshc, FrescoBot, Citation bot 1, Tkuvho, Trilobyte fossil, EmausBot, Niel Shell, ZéroBot, Jcimpric, Helpful Pixie Bot, Spaetzle, Larsborn, Giorgi Balakhadze, Rolf h nelson and Anonymous: 25 • Arithmetic and geometric Frobenius Source: https://en.wikipedia.org/wiki/Arithmetic_and_geometric_Frobenius?oldid=459057897 Contributors: Charles Matthews, Chtito, SmackBot, Bluebot, Headbomb, Jakob.scholbach, Niceguyedc and Erik9bot • Arithmetic dynamics Source: https://en.wikipedia.org/wiki/Arithmetic_dynamics?oldid=672615585 Contributors: Michael Hardy, Giftlite, Bgwhite, Siddhant, JosephSilverman, Welsh, SmackBot, Adam majewski, JohnBlackburne, Delaszk, Yobot, Charvest, Helpful Pixie Bot, CitationCleanerBot, Brad7777, Deltahedron, Enyokoyama and Anonymous: 4 • Artin L-function Source: https://en.wikipedia.org/wiki/Artin_L-function?oldid=644519466 Contributors: Zundark, Michael Hardy, Samw, Charles Matthews, Giftlite, Bender235, EmilJ, GregorB, BD2412, R.e.b., Mathbot, Masnevets, Doetoe, SmackBot, Bluebot, Nbarth, Robertwb, Headbomb, RobHar, Yill577, David Eppstein, LokiClock, Hesam7, Svick, Addbot, Yobot, AnomieBOT, LucienBOT, Citation bot 1, Trappist the monk, PoeticVerse, Helpful Pixie Bot, Enyokoyama, Spectral sequence, K9re11 and Anonymous: 12 • Artin reciprocity law Source: https://en.wikipedia.org/wiki/Artin_reciprocity_law?oldid=650297187 Contributors: Zundark, Michael Hardy, Charles Matthews, Gauge, EmilJ, MarSch, R.e.b., SmackBot, Headbomb, RobHar, Dugwiki, VectorPosse, David Eppstein, Arcfrk, Stca74, Cberry01, Ncsinger, Marc van Leeuwen, Addbot, Yobot, AnomieBOT, Ringspectrum, XPEHOPE3, Puzl bustr, DanielConstantinMayer, Deltahedron, Spectral sequence, Mark viking, GreenKeeper17 and Anonymous: 10 • Artin transfer (group theory) Source: https://en.wikipedia.org/wiki/Artin_transfer_(group_theory)?oldid=671388154 Contributors: Michael Hardy, Bearcat, Rjwilmsi, R.e.b., Magioladitis, AnomieBOT, DanielConstantinMayer, GreenKeeper17, Elysion, IsabelleSummer and Anonymous: 1 • Artin’s conjecture on primitive roots Source: https://en.wikipedia.org/wiki/Artin’{}s_conjecture_on_primitive_roots?oldid=665710839 Contributors: Michael Hardy, Charles Matthews, Hyacinth, Lowellian, Giftlite, EmilJ, Bobo192, GregorB, Mathbot, Maxal, X42bn6, Lenthe, DynaBlast, SmackBot, Jeppesn, Gutworth, CRGreathouse, Ntsimp, Mon4, PipepBot, Virginia-American, Addbot, Super duper jimbo, Ptbotgourou, David Brink, Citation bot, Citation bot 1, Dinamik-bot, RjwilmsiBot, Zatrp, K9re11 and Anonymous: 12 • Biquadratic field Source: https://en.wikipedia.org/wiki/Biquadratic_field?oldid=656906322 Contributors: Charles Matthews, SmackBot, Bluebot, Sadads, Headbomb, RobHar, Erik9bot, Midas02 and Anonymous: 1 • Brauer group Source: https://en.wikipedia.org/wiki/Brauer_group?oldid=651551646 Contributors: TakuyaMurata, Charles Matthews, MathMartin, Tobias Bergemann, Giftlite, Fropuff, Waltpohl, Vivacissamamente, Gauge, Spoon!, Rjwilmsi, R.e.b., Nbarth, Thijs!bot, Vanish2, Hesam7, Arcfrk, Lightmouse, JackSchmidt, Lkadison, MystBot, Addbot, YogaYak, Yobot, Ringspectrum, BenzolBot, Darij, ChrisGualtieri, Deltahedron, Spectral sequence, Mark viking and Anonymous: 16 • Brauer–Wall group Source: https://en.wikipedia.org/wiki/Brauer%E2%80%93Wall_group?oldid=622048703 Contributors: Fropuff, R.e.b., Headbomb, David Eppstein, Yobot, ChrisGualtieri, Deltahedron, K9re11 and Anonymous: 1 • Brumer–Stark conjecture Source: https://en.wikipedia.org/wiki/Brumer%E2%80%93Stark_conjecture?oldid=671204311 Contributors: Zundark, Michael Hardy, Gabbe, Charles Matthews, Giftlite, Bender235, CRGreathouse, RobHar, B2smith, UnCatBot, Ozob, Yobot, Kilom691, Deltahedron, Spectral sequence and Anonymous: 6 • Carlitz exponential Source: https://en.wikipedia.org/wiki/Carlitz_exponential?oldid=566357550 Contributors: Michael Hardy, Jitse Niesen, RobHar, Addbot, AnomieBOT, Bellerophon, ArticlesForCreationBot, Escspeed and Anonymous: 2 • Characteristic (algebra) Source: https://en.wikipedia.org/wiki/Characteristic_(algebra)?oldid=666775127 Contributors: AxelBoldt, LC~enwiki, Bryan Derksen, Zundark, Michael Hardy, Revolver, Charles Matthews, Steinsky, JensMueller, Moink, Jleedev, Tosha, Giftlite, BenFrantzDale, Fropuff, Dratman, Rpchase, Barnaby dawson, Guanabot, Pjacobi, MuDavid, Paul August, Rgdboer, Emvee~enwiki, Joriki, LOL, Graham87, Salix alba, YurikBot, Pred, GrinBot~enwiki, Eskimbot, Mhss, Nbarth, Lesnail, Vina-iwbot~enwiki, Xtv, Satori Son, W3asal, RobHar, JAnDbot, David Eppstein, JoergenB, STBot, Sigmundur, VolkovBot, LokiClock, TXiKiBoT, Kyle Pena, Finlux, SieBot, YonaBot, SimonTrew, Malatinszky, Anchor Link Bot, ArdClose, Niceguyedc, BOTarate, Addbot, Zorrobot, Luckas-bot, AnomieBOT, Rckrone, Javasava, ZéroBot, Offsure, Quondum, BG19bot, RuHouse'ls, ChrisGualtieri, GeoffreyT2000 and Anonymous: 25 • Class field theory Source: https://en.wikipedia.org/wiki/Class_field_theory?oldid=662858160 Contributors: Zundark, Michael Hardy, TakuyaMurata, CatherineMunro, Charles Matthews, Tobias Bergemann, Giftlite, Gene Ward Smith, Zhen Lin, Johnflux, Vivacissamamente, Gauge, Mdd, Tfz, R.e.b., Arthur Rubin, SmackBot, ProveIt, Medude24, Vina-iwbot~enwiki, Waggers, Valoem, Headbomb,
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CHAPTER 244. ZAHLBERICHT
Rnealh, RobHar, Dugwiki, T0, Jakob.scholbach, JohnBlackburne, TXiKiBoT, Pjoef, JackSchmidt, Nn123645, DeaconJohnFairfax, Ncsinger, Addbot, LaaknorBot, Yobot, Omnipaedista, Eugene-elgato, Citation bot 1, Mathsgg, RjwilmsiBot, EmausBot, Otaria, Chrangers, Pootle22, Shotaro Makisumi, Brad7777, Abclassfield, Deltahedron, Out shuffle, Enyokoyama, Harder133, Brirush, Hamoudafg, Ulam1988, KasparBot and Anonymous: 29 • Class formation Source: https://en.wikipedia.org/wiki/Class_formation?oldid=668981278 Contributors: Michael Hardy, Charles Matthews, Giftlite, Rich Farmbrough, EmilJ, Rjwilmsi, R.e.b., SmackBot, Jim.belk, CRGreathouse, WLior, Myasuda, Thijs!bot, Headbomb, RobHar, Forgetfulfunctor, Jakob.scholbach, Hesam7, Addbot, Yobot, Citation bot, Trappist the monk, Frietjes and Anonymous: 5 • Class number formula Source: https://en.wikipedia.org/wiki/Class_number_formula?oldid=605434435 Contributors: Michael Hardy, Charles Matthews, Icairns, R.e.b., User24, Jtwdog, CRGreathouse, RobHar, Magioladitis, Vanish2, EagleFan, GirasoleDE, SieBot, Addbot, Uncia, Yobot, Citation bot, RobinK, 777sms, Bineapple, Helpful Pixie Bot, Cjsh716, Enyokoyama, Jsinick and Anonymous: 11 • Class number problem Source: https://en.wikipedia.org/wiki/Class_number_problem?oldid=663862451 Contributors: Michael Hardy, Charles Matthews, MathMartin, Giftlite, Joseph Myers, Vivacissamamente, Sixpence, Gauge, Mlessard, Chenxlee, Rjwilmsi, R.e.b., Algebraist, RussBot, Eskimbot, Nbarth, Thepizzaking, Myasuda, Ntsimp, Headbomb, Woody, RobHar, Vanish2, Turtlens, TheSeven, LokiClock, Hesam7, Virginia-American, Addbot, Roentgenium111, DOI bot, Yobot, Citation bot, MathHisSci, Citation bot 1, Helpful Pixie Bot, Enyokoyama, Monkbot, Alpoge and Anonymous: 7 • CM-field Source: https://en.wikipedia.org/wiki/CM-field?oldid=655154640 Contributors: Charles Matthews, Giftlite, R.e.b., SmackBot, CRGreathouse, RobHar, David Eppstein, Omerks, Roentgenium111, Yobot, Deltahedron, Spectral sequence, Brirush, JohnAu2000 and Anonymous: 4 • Compatible system of ℓ-adic representations Source: https://en.wikipedia.org/wiki/Compatible_system_of_%E2%84%93-adic_representations? oldid=635096456 Contributors: Amalthea, CRGreathouse, Headbomb, RobHar, CharlotteWebb, Magioladitis, Helpful Pixie Bot and Brirush • Complete field Source: https://en.wikipedia.org/wiki/Complete_field?oldid=545546751 Contributors: RobHar, JackSchmidt, Addbot, Obersachsebot, Ringspectrum and Qetuth • Complex multiplication Source: https://en.wikipedia.org/wiki/Complex_multiplication?oldid=665011200 Contributors: Edward, Michael Hardy, Dominus, Eric119, Ahoerstemeier, Charles Matthews, Phil Boswell, Giftlite, Gauge, MFH, Rjwilmsi, Eubot, Dmharvey, Lenthe, Seb35, Hv, SmackBot, Gutworth, RyanEberhart, Mksword, Titus III, Vanisaac, CBM, RobHar, Dugwiki, Magioladitis, Arcfrk, Marc van Leeuwen, Addbot, Yobot, Freebirth Toad, Omnipaedista, D.Lazard, Helpful Pixie Bot, ChrisGualtieri, Deltahedron, Enyokoyama, Citizentoad, Escspeed, Monkbot and Anonymous: 9 • Composite field (mathematics) Source: https://en.wikipedia.org/wiki/Composite_field_(mathematics)?oldid=468896440 Contributors: Woohookitty, SmackBot, Afitzy, Pascal.Tesson, Alaibot, Vanish2, Luccul and Qetuth • Conductor (class field theory) Source: https://en.wikipedia.org/wiki/Conductor_(class_field_theory)?oldid=657216857 Contributors: Michael Hardy, Salix alba, Valoem, Headbomb, RobHar, Roentgenium111, 4meter4, Helpful Pixie Bot, Deltahedron, Spectral sequence and Anonymous: 1 • Conductor of an abelian variety Source: https://en.wikipedia.org/wiki/Conductor_of_an_abelian_variety?oldid=657216455 Contributors: Zundark, Michael Hardy, Eric119, Giftlite, Valoem, RobHar, Vanish2, David Eppstein, WikHead, Yobot, Citation bot, Citation bot 1, Uni.Liu, Monkbot and Anonymous: 1 • Conductor-discriminant formula Source: https://en.wikipedia.org/wiki/Conductor-discriminant_formula?oldid=563011457 Contributors: Michael Hardy, R.e.b., Headbomb, RobHar, CharlesGillingham, Kilom691, Ringspectrum and Anonymous: 1 • Conjugate element (field theory) Source: https://en.wikipedia.org/wiki/Conjugate_element_(field_theory)?oldid=656906168 Contributors: Edward, Michael Hardy, GABaker, Charles Matthews, Gubbubu, EmilJ, Oleg Alexandrov, Maxal, SmackBot, Skapur, LokiClock, JackSchmidt, Addbot, Isheden, Erik9bot, VAP+VYK, FoxBot, EmausBot, ZéroBot, Midas02, Wcherowi, Qetuth, Knocksheelan, Maimonid and Anonymous: 3 • Cubic field Source: https://en.wikipedia.org/wiki/Cubic_field?oldid=661376542 Contributors: Michael Hardy, Gabbe, TakuyaMurata, GTBacchus, Bender235, Chenxlee, Rjwilmsi, CBM, Headbomb, RobHar, David Eppstein, GirasoleDE, Bagworm, Cst17, 4meter4, DanielConstantinMayer, Anrnusna, Teddyktchan and Anonymous: 8 • Cubic reciprocity Source: https://en.wikipedia.org/wiki/Cubic_reciprocity?oldid=637904174 Contributors: Michael Hardy, SebastianHelm, Charles Matthews, Giftlite, Pmanderson, Nroets, Maxal, Sodin, RussBot, Chris the speller, DRLB, Lambiam, Dirk math, Jim.belk, LDH, CRGreathouse, Gyopi, ShelfSkewed, Thijs!bot, RobHar, Magioladitis, Vanish2, David Eppstein, STBot, Phe-bot, Virginia-American, Addbot, Lunae, Citation bot 1, Tal physdancer, Helpful Pixie Bot, BattyBot and Anonymous: 13 • Cyclotomic character Source: https://en.wikipedia.org/wiki/Cyclotomic_character?oldid=564525345 Contributors: Michael Hardy, DemonThing, Amalthea, RobHar, CharlotteWebb, JackSchmidt, Roentgenium111, FrescoBot and Anonymous: 5 • Cyclotomic field Source: https://en.wikipedia.org/wiki/Cyclotomic_field?oldid=670677094 Contributors: AxelBoldt, Dominus, TakuyaMurata, Charles Matthews, Dmadeo, (:Julien:), Giftlite, Inter, Mike40033, Cmprince, R.e.b., SmackBot, Incnis Mrsi, Daqu, RJChapman, CapitalR, CRGreathouse, Gyopi, Thijs!bot, GromXXVII, Yill577, Magioladitis, Vanish2, JamesBWatson, Policron, DavidCBryant, Arcfrk, Omerks, FractalFusion, Alexbot, PixelBot, Addbot, DavidLHarden, SpBot, PV=nRT, Luckas-bot, Yobot, Amirobot, DemocraticLuntz, Ringspectrum, LucienBOT, Vicenarian, Lapinskicho, Stephan Alexander Spahn, Spectral sequence, Tango303 and Anonymous: 9 • Cyclotomic unit Source: https://en.wikipedia.org/wiki/Cyclotomic_unit?oldid=595024007 Contributors: R.e.b., JosephSilverman, SmackBot, Jim.belk, RJChapman, RobHar, Vanish2, TXiKiBoT, Addbot, RjwilmsiBot, John of Reading, Uni.Liu, Boodlepounce, Deltahedron, Sanfoh and Anonymous: 1 • Dedekind domain Source: https://en.wikipedia.org/wiki/Dedekind_domain?oldid=664905510 Contributors: AxelBoldt, Zundark, Michael Hardy, Alodyne, TakuyaMurata, Ciphergoth, Charles Matthews, MathMartin, Giftlite, Gene Ward Smith, Waltpohl, Ojl, Vivacissamamente, 4pq1injbok, HasharBot~enwiki, Oleg Alexandrov, Rjwilmsi, Wavelength, Crasshopper, Zhw, SmackBot, Bluebot, Nbarth, BlackFingolfin, Tesseran, Zero sharp, Hebrides, Magioladitis, Policron, VolkovBot, Plclark, Omerks, SieBot, ToePeu.bot, JackSchmidt, Addbot, Roentgenium111, Frobitz, ThanosPapaioannou, Luckas-bot, Yobot, Phiech, Citation bot, Freebirth Toad, LucienBOT, Citation bot 1, Darij, MastiBot, Akerans, Anita5192, CitationCleanerBot, Alexjbest, Spectral sequence, Mark viking, Mcshantz and Anonymous: 32
244.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
727
• Dedekind zeta function Source: https://en.wikipedia.org/wiki/Dedekind_zeta_function?oldid=622044906 Contributors: Kku, Charles Matthews, Giftlite, Phe, Alexrexpvt, Pt, Eric Kvaalen, Linas, Chenxlee, R.e.b., John Baez, Mathbot, Reedy, Daqu, Anguis, Robertwb, Headbomb, RobHar, Vanish2, STBotD, TXiKiBoT, Hesam7, Omerks, GirasoleDE, DumZiBoT, Virginia-American, Addbot, Citation bot, FrescoBot, RedBot, 4meter4, ZéroBot, Helpful Pixie Bot, Spectral sequence and Anonymous: 10 • Degree of a field extension Source: https://en.wikipedia.org/wiki/Degree_of_a_field_extension?oldid=667789846 Contributors: AxelBoldt, MathMartin, Goochelaar, Arthena, Oleg Alexandrov, Dmharvey, Mathaxiom~enwiki, Bluebot, Jim.belk, RobHar, Ideal gas equation, SuperHamster, Addbot, Glane23, Citation bot, Devingragg, EmausBot, WikitanvirBot, 28bot, Braincricket, Mark viking and Anonymous: 5 • Different ideal Source: https://en.wikipedia.org/wiki/Different_ideal?oldid=657216162 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, Giftlite, Gauge, Chenxlee, Salix alba, Bluebot, Valoem, RobHar, Vanish2, GirasoleDE, Addbot, Queenmomcat, Eumolpo, El Caro, Ringspectrum, Citation bot 1, PigFlu Oink, CitationCleanerBot, Deltahedron, Spectral sequence and Anonymous: 6 • Differential Galois theory Source: https://en.wikipedia.org/wiki/Differential_Galois_theory?oldid=638925369 Contributors: Michael Hardy, GTBacchus, Charles Matthews, Wikiborg, Fredrik, Giftlite, Lethe, Jason Quinn, Shanes, EmilJ, Army1987, Woodstone, Oleg Alexandrov, Joriki, Linas, Shreevatsa, XaosBits, Rjwilmsi, R.e.b., [email protected], Pmdboi, SmackBot, Nbarth, Emurphy42, Jim.belk, Goens, Paddles, Xantharius, Headbomb, Warut, Squids and Chips, VolkovBot, Strcpy~enwiki, Mike4ty4, Gamesguru2, BOTarate, Addbot, Ettrig, Unara, UncleSpacker, Galoa2804~enwiki, Levochik, CaroleHenson, Spectral sequence, Brirush and Anonymous: 19 • Discrete valuation Source: https://en.wikipedia.org/wiki/Discrete_valuation?oldid=649523263 Contributors: AxelBoldt, Michael Hardy, Charles Matthews, Mat cross, Margosbot~enwiki, SmackBot, Incnis Mrsi, Headbomb, RobHar, Daniele.tampieri, Addbot, Erik9bot, Le schtroumpf, Helpful Pixie Bot, K9re11 and Anonymous: 5 • Discriminant Source: https://en.wikipedia.org/wiki/Discriminant?oldid=672777882 Contributors: AxelBoldt, Zundark, Michael Hardy, Wshun, Den fjättrade ankan~enwiki, Charles Matthews, Dino, Dysprosia, Robbot, Jwpurple, Georg Muntingh, Tosha, Giftlite, BenFrantzDale, Calmofthestorm, Discospinster, Paul August, DcoetzeeBot~enwiki, Bender235, Gauge, Erik456, Gary, Trhaynes, Rodii, Mindmatrix, CyrilleDunant, Marudubshinki, Salix alba, FlaBot, Mathbot, McAusten, Petrvs~enwiki, Glenn L, Imnotminkus, King of Hearts, Chobot, Dylan Thurston, DVdm, Dmharvey, Gene.arboit, Michael Slone, Danuthaiduc, Sir48, Zephalis, Hirak 99, LarryLACa, SmackBot, Thunderboltz, Bluebot, Nbarth, Hdgcfcf, Akriasas, Coredesat, Asyndeton, Robertwb, Igoldste, Valoem, 345Kai, MaxEnt, Doctormatt, SpK, Thijs!bot, Pmagyar, Knakts, RobHar, Eleuther, Wang ty87916, Lklundin, 01001, VoABot II, Xiahou, VolkovBot, Hellohihihihi, Davidsevilla, Jobu0101, Anonymous Dissident, Arcfrk, SieBot, Dawn Bard, Cwkmail, Svick, The sunder king, DavidHobby, ClueBot, DionysosProteus, Southrop, Supertouch, Mild Bill Hiccup, Lexa122, Excirial, Alexbot, Estirabot, Alexey Muranov, Ncsinger, Qwfp, RMFan1, Gonfer, Deineka, Addbot, MrOllie, Sanawon, Tide rolls, Ajstern, Zorrobot, Luckas-bot, THEN WHO WAS PHONE?, Halothane, Naderra, AnomieBOT, Dans595, Unara, Materialscientist, DannyAsher, Xqbot, Σ, Jujutacular, Mean as custard, RjwilmsiBot, Rhythms06, Jowa fan, EmausBot, Tommy2010, TuHan-Bot, David2121~enwiki, Chharvey, Sbealing, D.Lazard, Electron0511, ClueBot NG, Chester Markel, Johnma97, Helpful Pixie Bot, MusikAnimal, Comfr, ChrisGualtieri, Deltahedron, SBareSSomErMig, Pokajanje, SucreRouge, Weux082690, Fowlslegs, GeoffreyT2000, Loraof, Jerry08baddog, Batman’s butler and Anonymous: 122 • Discriminant of an algebraic number field Source: https://en.wikipedia.org/wiki/Discriminant_of_an_algebraic_number_field?oldid= 651899007 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, Chuunen Baka, Giftlite, D6, Gauge, Rjwilmsi, Gaius Cornelius, Light current, Jim.belk, CRGreathouse, Headbomb, RobHar, Vanish2, Hesam7, Ferengi, Andreas Carter, GirasoleDE, CharlesGillingham, Addbot, Citation bot, Ringspectrum, Louperibot, Citation bot 1, 4meter4, Helpful Pixie Bot, CitationCleanerBot, Deltahedron, Spectral sequence, A.entropy, CaterinaSchneider and Anonymous: 11 • Drinfeld module Source: https://en.wikipedia.org/wiki/Drinfeld_module?oldid=612109019 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, Altenmann, Giftlite, Mboverload, R.e.b., SmackBot, 'Ff'lo, CRGreathouse, Thijs!bot, Headbomb, Wlod, PeterStJohn, David Eppstein, Gwern, Addbot, Yobot, Woodchiu, Verhoek, Deltahedron, Escspeed, Dirk Basson and Anonymous: 12 • Dual basis in a field extension Source: https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension?oldid=644686663 Contributors: Charles Matthews, Gene Ward Smith, CryptoDerk, ArnoldReinhold, SmackBot, Ideal gas equation, Niceguyedc, Erik9bot, Brad7777, LimeyCinema1960 and Anonymous: 1 • Eisenstein reciprocity Source: https://en.wikipedia.org/wiki/Eisenstein_reciprocity?oldid=650836555 Contributors: Giftlite, R.e.b., Virginia-American, Yobot, Trappist the monk, Josve05a, Maschen, TuxLibNit and Anonymous: 2 • Eisenstein sum Source: https://en.wikipedia.org/wiki/Eisenstein_sum?oldid=672064943 Contributors: R.e.b., Bgwhite, David Eppstein, Yobot, Deltahedron and K9re11 • Eisenstein’s criterion Source: https://en.wikipedia.org/wiki/Eisenstein’{}s_criterion?oldid=657213897 Contributors: Michael Hardy, Revolver, Charles Matthews, Dysprosia, Ktotam, Marc Venot, Giftlite, Brequinda, Klemen Kocjancic, Gauge, El C, EmilJ, Obradovic Goran, Oleg Alexandrov, Stolee, Ryan Reich, Chenxlee, Rjwilmsi, YurikBot, Michael Slone, Danlaycock, Matikkapoika~enwiki, KnightRider~enwiki, Reedy, Melchoir, Lambiam, Valoem, Wafulz, Stebulus, Mon4, Thijs!bot, LaGrange, RobHar, WelcomeToDie, SpecZ, Albmont, Steve Checkoway, SieBot, AlphaPyro, PipepBot, Marc van Leeuwen, Addbot, Whyiseverythingused, Luckas-bot, Yobot, Calle, Anythingapplied, Cpryby, Duoduoduo, EmausBot, ZéroBot, D.Lazard, Nicelbole, Spectral sequence, Parkerf and Anonymous: 28 • Elementary number Source: https://en.wikipedia.org/wiki/Elementary_number?oldid=557222696 Contributors: Michael Hardy, Bearcat, Mathbot, CRGreathouse and Dekart • Elliptic Gauss sum Source: https://en.wikipedia.org/wiki/Elliptic_Gauss_sum?oldid=626848206 Contributors: Michael Hardy, Giftlite, R.e.b., Raulshc and Trappist the monk • Elliptic unit Source: https://en.wikipedia.org/wiki/Elliptic_unit?oldid=650901686 Contributors: Oleg Alexandrov, Rjwilmsi, R.e.b., TexasAndroid, Headbomb, RobHar, Vanish2, Deltahedron, K9re11 and Anonymous: 1 • Embedding problem Source: https://en.wikipedia.org/wiki/Embedding_problem?oldid=639922602 Contributors: Michael Hardy, Tobias Bergemann, RussBot, Ntsimp, Alaibot, RebelRobot, Barylior, Geometry guy, Yobot, Vshmath, Playmobilonhishorse, Spectral sequence, Mark viking, Mohd.qat and Anonymous: 4 • Equally spaced polynomial Source: https://en.wikipedia.org/wiki/Equally_spaced_polynomial?oldid=662175323 Contributors: Charles Matthews, Giftlite, CryptoDerk, Phantomsteve, Geach, Vanish2, Mogism and Anonymous: 1
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CHAPTER 244. ZAHLBERICHT
• Equivariant L-function Source: https://en.wikipedia.org/wiki/Equivariant_L-function?oldid=612113070 Contributors: Michael Hardy, Bender235, RobHar, David Eppstein, R'n'B, JL-Bot, B2smith, UnCatBot, AnomieBOT, Brad7777 and Deltahedron • Euclidean field Source: https://en.wikipedia.org/wiki/Euclidean_field?oldid=644748101 Contributors: Michael Hardy, R.e.b., SmackBot, Sullivan.t.j, Mild Bill Hiccup, Addbot, Luckas-bot, Omnipaedista, D.Lazard, Snapelex, Deltahedron, Spectral sequence, Mark viking and Anonymous: 2 • Euler system Source: https://en.wikipedia.org/wiki/Euler_system?oldid=622301335 Contributors: RTC, Charles Matthews, Giftlite, Rjwilmsi, R.e.b., FlaBot, Dmharvey, Headbomb, RobHar, STBotD, Kyle the bot, Astronautb, Addbot, Yobot, Imbalzanog, Lagelspeil, RjwilmsiBot, Boodlepounce, Deltahedron, K9re11 and Anonymous: 2 • Explicit reciprocity law Source: https://en.wikipedia.org/wiki/Explicit_reciprocity_law?oldid=626848245 Contributors: Michael Hardy, TakuyaMurata, Rjwilmsi, R.e.b., Yobot, Trappist the monk, RjwilmsiBot, Deltahedron, Spectral sequence and Anrnusna • Exponential field Source: https://en.wikipedia.org/wiki/Exponential_field?oldid=653907897 Contributors: Sanxiyn, Tobias Bergemann, Chenxlee, Nbarth, Headbomb, LokiClock, Joule36e5, Wk99, Quondum, Gergely.Szekely and Anonymous: 1 • Exponentially closed field Source: https://en.wikipedia.org/wiki/Exponentially_closed_field?oldid=607875153 Contributors: Rjwilmsi, Joule36e5, Gergely.Szekely, Brad7777 and Deltahedron • Extension and contraction of ideals Source: https://en.wikipedia.org/wiki/Extension_and_contraction_of_ideals?oldid=666230519 Contributors: Revolver, Charles Matthews, Paul August, SmackBot, Mathemajor, FrescoBot, Fraulein451, ChrisGualtieri and Anonymous: 4 • Field (mathematics) Source: https://en.wikipedia.org/wiki/Field_(mathematics)?oldid=670186165 Contributors: AxelBoldt, Bryan Derksen, Zundark, The Anome, Andre Engels, Josh Grosse, XJaM, Toby Bartels, Miguel~enwiki, Lir, Patrick, Michael Hardy, Wshun, DIG~enwiki, TakuyaMurata, Karada, Looxix~enwiki, Rossami, Andres, Loren Rosen, Revolver, RodC, Dysprosia, Jitse Niesen, Prumpf, Tero~enwiki, Phys, Philopp, R3m0t, Jmabel, Mattblack82, MathMartin, P0lyglut, Wikibot, Tobias Bergemann, Unfree, Marc Venot, Giftlite, Highlandwolf, Gene Ward Smith, Lethe, Zigger, Fropuff, Millerc, Waltpohl, Python eggs, Gubbubu, CSTAR, Pmanderson, Barnaby dawson, PhotoBox, Mormegil, Jørgen Friis Bak, Discospinster, Guanabot, Sperling, Paul August, Zaslav, Elwikipedista~enwiki, El C, Rgdboer, EmilJ, Touriste, Army1987, Giraffedata, Obradovic Goran, OoberMick, Msh210, Mlm42, Olegalexandrov, RJFJR, Oleg Alexandrov, Woohookitty, Linas, Arneth, Bkkbrad, Hypercube~enwiki, MarkTempeit, Damicatz, MFH, Isnow, Palica, Graham87, FreplySpang, Chenxlee, Josh Parris, Rjwilmsi, Hiberniantears, Salix alba, R.e.b., FlaBot, Codazzi~enwiki, Jrtayloriv, R160K, Chobot, Abu Amaal, Algebraist, Wavelength, Dmesg, Eraserhead1, Hairy Dude, KSmrq, Grubber, Archelon, Rintrah, Rat144, Rick Norwood, Trovatore, DYLAN LENNON~enwiki, Crasshopper, RaSten, DavidHouse~enwiki, Mgnbar, Children of the dragon, SmackBot, Mmernex, Melchoir, Gilliam, Nbarth, Charlotte Hobbs, Lesnail, Cybercobra, Acepectif, Slawekk, Bidabadi~enwiki, Lambiam, Jim.belk, Schildt.a, Mets501, DabMachine, Rschwieb, WAREL, Newone, Vaughan Pratt, CRGreathouse, Kupirijo, Tiphareth, DEWEY, Eulerianpath, Pedro Fonini, Goldencako, BobNiichel, Xantharius, KLIP~enwiki, JLISP, Headbomb, RobHar, Nick Number, Turgidson, Kprateek88, Martinkunev, Magioladitis, Bongwarrior, VoABot II, JamesBWatson, Jakob.scholbach, SwiftBot, Catgut, Lukeaw, MORI, Cpiral, Maproom, Gombang, Policron, Barylior, Umarekawari, LokiClock, Red Act, Anonymous Dissident, Hesam7, Joeldl, Dave703, Zermalo, Shellgirl, Cwkmail, Soler97, JackSchmidt, Jorgen W, Anchor Link Bot, Willy, your mate, Oekaki, UKe-CH, ClueBot, Mild Bill Hiccup, Tcklein, Niceguyedc, He7d3r, Bender2k14, Squirreljml, Palnot, ZooFari, Addbot, Gabriele ricci, Download, Unzerlegbarkeit, Cesiumfrog, Yobot, Ht686rg90, TaBOT-zerem, Zagothal, AnomieBOT, DSisyphBot, Depassp, Danielschreiber, MegaMouthBolt123, Point-set topologist, Charvest, KirarinSnow, FrescoBot, Mjmarkowitz, RandomDSdevel, Ebony Jackson, D stankov, Girish.ponkiya2007, Kunle102, DASHBot, Sedrikov, Tom.kemp90, Tommy2010, Wikipelli, Shishir332, Lfrazier11, Quondum, D.Lazard, JimMeiss, ClueBot NG, Ankur1vi, Wcherowi, Frietjes, MerlIwBot, Helpful Pixie Bot, !mcbloobyenstein!!, Or elharar, Fabio.nsantos, Rjs.swarnkar, Topgraph28, Deltahedron, Sanipriya, GigaGerard, CsDix, YiFeiBot, Teddyktchan, GeoffreyT2000, Charlotte Aryanne, Vluczkow and Anonymous: 133 • Field extension Source: https://en.wikipedia.org/wiki/Field_extension?oldid=655247312 Contributors: AxelBoldt, Zundark, Edward, TakuyaMurata, Daran, Naddy, Lowellian, MathMartin, Giftlite, Fropuff, Mazi, El C, Bookofjude, EmilJ, Oleg Alexandrov, Marudubshinki, Graham87, FlaBot, YurikBot, Dmharvey, Mathaxiom~enwiki, Grubber, KnightRider~enwiki, SmackBot, Nbarth, Foxjwill, Ewjw, Jim.belk, Madmath789, CRGreathouse, Thijs!bot, RobHar, Escarbot, JAnDbot, Magioladitis, David Eppstein, Cpiral, Policron, STBotD, PerezTerron, TXiKiBoT, Don4of4, AlleborgoBot, Cwkmail, JackSchmidt, Yasmar, Ideal gas equation, Mpd1989, Alexbot, He7d3r, Addbot, PV=nRT, Ptbotgourou, Calle, Xqbot, Point-set topologist, Uuo, Sławomir Biały, Vanished user fijtji34toksdcknqrjn54yoimascj, Cenkner, YFdyh-bot and Anonymous: 46 • Field norm Source: https://en.wikipedia.org/wiki/Field_norm?oldid=672065115 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, MathMartin, Rgdboer, Burn, Joriki, Dmharvey, SmackBot, Adamp, MvH, JoshuaZ, CRGreathouse, Thijs!bot, RobHar, WinBot, GromXXVII, David Eppstein, Aretakis, Malerin, AlleborgoBot, Addbot, Frobitz, Luckas-bot, Unara, Erik9bot, WikitanvirBot, Quondum, Wcherowi, Citizentoad and Anonymous: 10 • Field of fractions Source: https://en.wikipedia.org/wiki/Field_of_fractions?oldid=638300043 Contributors: AxelBoldt, Zundark, Michael Hardy, TakuyaMurata, GTBacchus, Ciphergoth, Loren Rosen, Charles Matthews, MathMartin, Tobias Bergemann, Giftlite, Fropuff, Vivacissamamente, Zaslav, Rgdboer, Arneth, Ruud Koot, Magidin, FlaBot, Régis Décamps, Iridescent, Zero sharp, Thijs!bot, RobHar, Escarbot, Salgueiro~enwiki, JamesBWatson, LokiClock, Kyle the bot, Egaida, Anonymous Dissident, SieBot, Cwkmail, Qwfp, Marc van Leeuwen, SpBot, Ozob, Zorrobot, Legobot, Luckas-bot, Yobot, Rubinbot, Xqbot, Isheden, Adam Dent, Erik9bot, FrescoBot, Ebony Jackson, ElNuevoEinstein, WikitanvirBot and Anonymous: 21 • Field trace Source: https://en.wikipedia.org/wiki/Field_trace?oldid=672065078 Contributors: Stevertigo, Michael Hardy, Charles Matthews, Gubbubu, Crisófilax, Dmharvey, SmackBot, Bluebot, Thijs!bot, JustAGal, RobHar, David Eppstein, Kyle the bot, Wikiisawesome, Addbot, Luckas-bot, Obersachsebot, Erik9bot, Trappist the monk, ChuispastonBot, Proz, Wcherowi, Deltahedron, Spectral sequence and Anonymous: 1 • Finite extensions of local fields Source: https://en.wikipedia.org/wiki/Finite_extensions_of_local_fields?oldid=659322263 Contributors: Michael Hardy, TakuyaMurata, Malcolma, David Eppstein, AvicBot, Helpful Pixie Bot, Qetuth, Deltahedron, Spectral sequence and Anonymous: 7 • Formal group Source: https://en.wikipedia.org/wiki/Formal_group?oldid=672793856 Contributors: Michael Hardy, Charles Matthews, Giftlite, Waltpohl, Gauge, Mdd, Rjwilmsi, R.e.b., Wavelength, JosephSilverman, SmackBot, Lhf, FelisLeo, SashatoBot, S.H.C., Michael Kinyon, Krasnoludek, Headbomb, Darklilac, David Eppstein, Gwern, Cardano~enwiki, Saibod, Brenont, JackSchmidt, Addbot, AnomieBOT, Udoh, Freebirth Toad, Tathagata84, HawaiianEarring, BG19bot, ChrisGualtieri, Deltahedron, Spectral sequence and Anonymous: 17
244.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
729
• Formally real field Source: https://en.wikipedia.org/wiki/Formally_real_field?oldid=639513455 Contributors: Charles Matthews, Aleph4, Giftlite, Gene Ward Smith, D6, ArnoldReinhold, VKokielov, Algebraist, YurikBot, Cheeser1, Kompik, Arthur Rubin, SmackBot, Texas Dervish, Sniffnoy, Oerjan, DesolateReality, Yobot, AnomieBOT, Erik9bot, Julian Birdbath, AvicAWB, Deltahedron and Anonymous: 5 • Fractional ideal Source: https://en.wikipedia.org/wiki/Fractional_ideal?oldid=665640506 Contributors: Zundark, TakuyaMurata, Ciphergoth, Charles Matthews, Gene Ward Smith, Waltpohl, Gauge, Gene Nygaard, Zhw, SmackBot, BlackFingolfin, Headbomb, RobHar, David Eppstein, DeaconJohnFairfax, Bogdanno, MystBot, Addbot, Expz, Luckas-bot, Erik9bot, Dewey process, WikitanvirBot, ZéroBot, Linus44 and Anonymous: 6 • Frobenius endomorphism Source: https://en.wikipedia.org/wiki/Frobenius_endomorphism?oldid=663185621 Contributors: Edward, TakuyaMurata, Charles Matthews, Gandalf61, MathMartin, Dmitri83, ComplexZeta, Giftlite, Gene Ward Smith, MathKnight, Gauge, HasharBot~enwiki, Ryan Reich, Salix alba, R.e.b., YurikBot, RyanEberhart, Thijs!bot, RobHar, Vanish2, David Eppstein, Arturj, VolkovBot, If brain hate soul?, Hesam7, Thehotelambush, JackSchmidt, Justin W Smith, Bender2k14, Estirabot, Marc van Leeuwen, Addbot, Roentgenium111, Ozob, Numbo3-bot, Luckas-bot, Yobot, Amirobot, AnomieBOT, AlreadyDone, Tdupu10000, Uni.Liu, YFdyh-bot, Enyokoyama, Spectral sequence, Danneks and Anonymous: 27 • Function field sieve Source: https://en.wikipedia.org/wiki/Function_field_sieve?oldid=573352800 Contributors: Michael Hardy, Ixfd64, Fivemack, Malcolma, Smartse, Somno, Charvest, Erik9bot, FrescoBot, Qetuth and Anonymous: 1 • Fundamental discriminant Source: https://en.wikipedia.org/wiki/Fundamental_discriminant?oldid=567927755 Contributors: Michael Hardy, TakuyaMurata, Simetrical, Rjwilmsi, Maxal, Chris the speller, Goldencako, Headbomb, RobHar, Vanish2, Kkilger, VirginiaAmerican, 4meter4, Helpful Pixie Bot and Anonymous: 3 • Fundamental theorem of algebra Source: https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra?oldid=669986818 Contributors: AxelBoldt, Mav, Zundark, XJaM, Toby Bartels, Michael Hardy, Wshun, Chinju, TakuyaMurata, Ahoerstemeier, Snoyes, AugPi, Holger Blasum, Charles Matthews, Dfeuer, Dysprosia, Jitse Niesen, Doradus, Markhurd, Michael Larsen, Furrykef, Saltine, Jimbreed, Andy Fugard, Ortonmc, Daran, Robbot, Fredrik, Romanm, MathMartin, Rholton, Robinh, Dmn, Tobias Bergemann, Tosha, Giftlite, Arved, Gene Ward Smith, Lupin, Dratman, Nsh, Wmahan, Bob.v.R, Icairns, Zfr, Smimram, Rich Farmbrough, Syp, Pt, Rgdboer, Aude, EmilJ, Dtremenak, .:Ajvol:., Toh, Dfeldmann, Obradovic Goran, Helix84, LutzL, Jumbuck, Alansohn, Arthena, Greg Kuperberg, Oleg Alexandrov, Nuno Tavares, Woohookitty, Jacobolus, Jimbryho, Mpatel, Graham87, Li-sung, Rjwilmsi, Salix alba, Mike Segal, DonSiano, [email protected], Mathbot, Chobot, Algebraist, YurikBot, Jimp, Michael Slone, Hede2000, SoroSuub1, Archelon, Gaius Cornelius, Trovatore, Arthur Rubin, MathsIsFun, Paul D. Anderson, Zvika, Adam majewski, BeteNoir, Unyoyega, Bluebot, JCSantos, Blindsuperhero, Huddlebum, Alink, Gutworth, DHN-bot~enwiki, Hgrosser, Can't sleep, clown will eat me, LkNsngth, Cybercobra, Meni Rosenfeld, Henning Makholm, Jóna Þórunn, Bidabadi~enwiki, Qmwne235, Lambiam, Lakinekaki, PseudoSudo, JdH, Tawkerbot2, Xantharius, Thijs!bot, RobHar, BigJohnHenry, AntiVandalBot, Kartik.jayan, Skomorokh, Sjlain, Ttwo, Trumpet marietta 45750, Smcinerney, Policron, PMajer, Nxavar, Hesam7, Geometry guy, Randomblue, Nic bor, Paul Taylor, Vladkornea, ChandlerMapBot, DragonBot, Lunchscale, Tulcod, SilvonenBot, Shishir0610, Deineka, Addbot, WikiUserPedia, LaaknorBot, Abovechief, Yobot, Fraggle81, Charleswallingford, AnomieBOT, Citation bot, Drilnoth, Oxy86, FrescoBot, Spartan S58, Monamip, Citation bot 1, Darij, AstaBOTh15, Primalbeing, Philologer, EmausBot, Fly by Night, Clive tooth, Slawekb, Tolly4bolly, CitationCleanerBot, Bob o Shiska, Brad7777, Epsilonball, ChrisGualtieri, Schauspieler~enwiki, Vincenthuang75025, Dexbot, Jmalmira, Nigellwh, Czj67890, Oswaldosimone, Chris.mindas and Anonymous: 111 • Fundamental theorem of Galois theory Source: https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory?oldid=661644990 Contributors: Charles Matthews, Dfeuer, Dysprosia, MathMartin, Giftlite, HorsePunchKid, Icairns, Frau Holle, Bender235, YurikBot, Dmharvey, BeteNoir, Eskimbot, Cybercobra, MarkC77, MvH, Jim.belk, Jibbb, Hesam7, Geometry guy, Cwkmail, Bender2k14, BOTarate, Sandrobt, Addbot, Yobot, Slawekb, D.Lazard, Anita5192, Helpful Pixie Bot, Brad7777, Maimonid and Anonymous: 18 • Fundamental unit (number theory) Source: https://en.wikipedia.org/wiki/Fundamental_unit_(number_theory)?oldid=654798476 Contributors: Rjwilmsi, MarSch, Srleffler, RobHar, Vanish2, David Eppstein, Citation bot, DrilBot, D.Lazard, ChrisGualtieri and Anonymous: 3 • Galois cohomology Source: https://en.wikipedia.org/wiki/Galois_cohomology?oldid=661611323 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, Giftlite, Rich Farmbrough, Gauge, Oleg Alexandrov, Crasshopper, SmackBot, Silly rabbit, CRGreathouse, CBM, Headbomb, RobHar, Dugwiki, Jakob.scholbach, GirasoleDE, Alexbot, Wefdsf, Addbot, Deltahedron, Sunny97you and Anonymous: 5 • Galois extension Source: https://en.wikipedia.org/wiki/Galois_extension?oldid=661350960 Contributors: Edward, Charles Matthews, MathMartin, Fuelbottle, Giftlite, EmilJ, Mdd, Algebraist, Dmharvey, Greatal386, SmackBot, Eskimbot, Gutworth, RyanEberhart, MvH, Thijs!bot, RobHar, Etale, TomyDuby, Sigmundur, TXiKiBoT, Omerks, Dogah, Cwkmail, Ideal gas equation, Alexbot, Bender2k14, Addbot, Luckas-bot, Xqbot, Point-set topologist, RibotBOT, Erik9bot, Anita5192, Enyokoyama, Brirush and Anonymous: 9 • Galois module Source: https://en.wikipedia.org/wiki/Galois_module?oldid=657214357 Contributors: Charles Matthews, Vivacissamamente, Paul August, Oleg Alexandrov, R.e.b., Bluebot, Jwillbur, Valoem, CRGreathouse, RobHar, STBotD, VolkovBot, Synthebot, Phe-bot, Addbot, LilHelpa, MauritsBot, DSisyphBot, Lagelspeil, Helpful Pixie Bot, Spectral sequence, K9re11 and Anonymous: 4 • Generic polynomial Source: https://en.wikipedia.org/wiki/Generic_polynomial?oldid=657189696 Contributors: Michael Hardy, Charles Matthews, MathMartin, Gene Ward Smith, Gauge, Gimboid13, Jim.belk, Scott Tillinghast, Houston TX, MegaMergatroid, Joc87, Yobot, Teddyktchan and Anonymous: 3 • Genus character Source: https://en.wikipedia.org/wiki/Genus_character?oldid=626886248 Contributors: R.e.b., TexasAndroid, RobHar, David Eppstein, Trappist the monk and Jesse V. • Genus field Source: https://en.wikipedia.org/wiki/Genus_field?oldid=639154730 Contributors: Vanish2, David Eppstein, DGG, SporkBot, Deltahedron and Anonymous: 1 • Global field Source: https://en.wikipedia.org/wiki/Global_field?oldid=640982806 Contributors: Zundark, Michael Hardy, Ciphergoth, Charles Matthews, Michael Larsen, MathMartin, Giftlite, Gene Ward Smith, Vivacissamamente, R.e.b., RobertG, Mathbot, Margosbot~enwiki, YurikBot, Michael Slone, Dan131m, CRGreathouse, Ntsimp, Mattisse, Thijs!bot, Headbomb, RobHar, Vanish2, Yonidebot, Sapphic, לירן, Addbot, Lightbot, Xqbot, Ringspectrum, Citation bot 1, Trappist the monk, Thue Siegel Roth, MathKnight-at-TAU, Brirush, Wikid77b, K9re11 and Anonymous: 6
730
CHAPTER 244. ZAHLBERICHT
• Glossary of field theory Source: https://en.wikipedia.org/wiki/Glossary_of_field_theory?oldid=648502002 Contributors: AxelBoldt, Michael Hardy, Wshun, Dcljr, Loren Rosen, Charles Matthews, Dysprosia, Robbot, Fropuff, D6, Rich Farmbrough, Ruud Koot, Marudubshinki, Marco Streng, Dmharvey, SmackBot, Xyzzyplugh, Cydebot, Nick Number, STBot, CopyToWiktionaryBot, Barylior, Addbot, Bte99, Yobot, Erik9bot, Fred Gandt, CaroleHenson, Deltahedron and Anonymous: 8 • Golod–Shafarevich theorem Source: https://en.wikipedia.org/wiki/Golod%E2%80%93Shafarevich_theorem?oldid=642716083 Contributors: Michael Hardy, Charles Matthews, Academic Challenger, Giftlite, Bender235, R.e.b., Michael Slone, SmackBot, Mathsci, Yonatbe5, RobHar, Turgidson, David Eppstein, Geometry guy, JackSchmidt, Yobot, Citation bot, Ebony Jackson, Jackbarron, Brad7777, Deltahedron, K9re11 and Anonymous: 2 • Grothendieck’s Galois theory Source: https://en.wikipedia.org/wiki/Grothendieck’{}s_Galois_theory?oldid=608106891 Contributors: Michael Hardy, Charles Matthews, Fropuff, Vivacissamamente, Michael Slone, Stephenb, Gaius Cornelius, Ilmari Karonen, RonnieBrown, Jim.belk, TKilbourn, RobHar, Reddyuday, Makotoy, Addbot, Yobot, Helpful Pixie Bot, Brad7777, Monkbot and Anonymous: 6 • Ground field Source: https://en.wikipedia.org/wiki/Ground_field?oldid=670273683 Contributors: Michael Hardy, Charles Matthews, Robertgreer, DexDor, Brirush and K9re11 • Group cohomology Source: https://en.wikipedia.org/wiki/Group_cohomology?oldid=652017954 Contributors: AxelBoldt, Zundark, Michael Hardy, TakuyaMurata, Charles Matthews, Dysprosia, Robbot, RedWolf, Giftlite, MathKnight, MSGJ, Waltpohl, Vivacissamamente, Rich Farmbrough, Zaslav, Gauge, Quasicharacter, Owenjonesuk, Msh210, Oleg Alexandrov, Joriki, DealPete, Marudubshinki, Rjwilmsi, R.e.b., Wavelength, Tob~enwiki, Bluebot, Silly rabbit, Nbarth, LkNsngth, JarahE, Vanisaac, CRGreathouse, CBM, RobHar, Jakob.scholbach, Etale, David Eppstein, Maproom, Trumpet marietta 45750, Hesam7, Jaswenso, SieBot, JackSchmidt, CharlesGillingham, Nsk92, MystBot, Addbot, Roentgenium111, Expz, Yobot, KamikazeBot, Talkloud, Citation bot, Yimuyin, Cohomology, Citation bot 1, Tkuvho, Alexander Chervov, ElNuevoEinstein, Omarct1989, Trodemaster, Deltahedron, Spectral sequence, Mark viking and Anonymous: 43 • Grunwald–Wang theorem Source: https://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_theorem?oldid=647255548 Contributors: Michael Hardy, Giftlite, Rjwilmsi, R.e.b., Sodin, Jessesaurus, CRGreathouse, Headbomb, RobHar, T@nn, David Eppstein, Cerebellum, Sun Creator, Rror, Addbot, Yobot, David Brink, Ringspectrum, Ebony Jackson, Trappist the monk, ZéroBot, Brad7777, GreenKeeper17 and Anonymous: 1 • Hardy field Source: https://en.wikipedia.org/wiki/Hardy_field?oldid=621667763 Contributors: Michael Hardy, Chenxlee, Niceguyedc, SchreiberBike, CitationCleanerBot, Rocksandwaves, Ab konst, LimeyCinema1960 and Anonymous: 1 • Hasse invariant of an algebra Source: https://en.wikipedia.org/wiki/Hasse_invariant_of_an_algebra?oldid=668981208 Contributors: Michael Hardy, Myasuda, Ironholds, ChrisGualtieri, Deltahedron, Spectral sequence and K9re11 • Hasse norm theorem Source: https://en.wikipedia.org/wiki/Hasse_norm_theorem?oldid=666743613 Contributors: Charles Matthews, Gene Ward Smith, R.e.b., Mathbot, Sodin, SmackBot, BeteNoir, Dugwiki, Vanish2, Kyle the bot, Arcfrk, Addbot, Clarkcj12, Brad7777, K9re11 and Anonymous: 2 • Hasse principle Source: https://en.wikipedia.org/wiki/Hasse_principle?oldid=626891846 Contributors: The Anome, Michael Hardy, Charles Matthews, Dysprosia, Giftlite, Gene Ward Smith, Apyule, Btyner, Rjwilmsi, R.e.b., Arthur Rubin, Nbarth, JoshuaZ, Stephen B Streater, WAREL, Mon4, Kilva, Vanish2, David Eppstein, STBot, Hair Commodore, Policron, Addbot, DOI bot, David Brink, AnomieBOT, Citation bot, Citation bot 1, Trappist the monk, Helpful Pixie Bot, Spectral sequence, Anrnusna, Monkbot and Anonymous: 9 • Heegner number Source: https://en.wikipedia.org/wiki/Heegner_number?oldid=660825102 Contributors: XJaM, Michael Hardy, Silverfish, Charles Matthews, Jotomicron, Psychonaut, Gandalf61, MathMartin, Mattflaschen, Giftlite, Fropuff, Alison, Pmanderson, Blotwell, Msh210, Oleg Alexandrov, Jörg Knappen~enwiki, Chobot, Jimp, SmackBot, RDBury, Incnis Mrsi, Eskimbot, PrimeHunter, Nbarth, Matchups, Titus III, CRGreathouse, RobHar, JamesBWatson, Mouchoir le Souris, Gfis, VolkovBot, UKe-CH, Zetud, DumZiBoT, Addbot, Lightbot, Luckas-bot, Srich32977, DixonDBot, Mark Messer, EmausBot, Slawekb, L Kensington, Helpful Pixie Bot, MrBill3, ChrisGualtieri, BeaumontTaz, Monkbot, Naclis and Anonymous: 15 • Heegner point Source: https://en.wikipedia.org/wiki/Heegner_point?oldid=654355949 Contributors: Michael Hardy, Giftlite, R.e.b., SmackBot, Ntsimp, Headbomb, David Eppstein, BQN, ComtedeMonteCristo, Citation bot 1, Fred Gandt, Helpful Pixie Bot, Enyokoyama, Mat4free and Anonymous: 5 • Herbrand quotient Source: https://en.wikipedia.org/wiki/Herbrand_quotient?oldid=582461520 Contributors: Zundark, R.e.b., Gareth Jones, SmackBot, Schmiteye, CRGreathouse, RobHar, Quotient group, Helpful Pixie Bot, Deltahedron, Mark viking and Anonymous: 1 • Hermite’s problem Source: https://en.wikipedia.org/wiki/Hermite’{}s_problem?oldid=658576807 Contributors: Chenxlee, Wavelength, Myasuda, Qwyrxian, DynamoDegsy and Anonymous: 4 • Higher local field Source: https://en.wikipedia.org/wiki/Higher_local_field?oldid=663600742 Contributors: Michael Hardy, Giftlite, Wavelength, RobHar, LokiClock, Wilhelmina Will, AnomieBOT, FrescoBot, Helpful Pixie Bot, BG19bot, Tomoliver163, Igor Zhukov~enwiki, Deltahedron, Out shuffle, Dirk Basson and Anonymous: 6 • Hilbert class field Source: https://en.wikipedia.org/wiki/Hilbert_class_field?oldid=668652268 Contributors: Zundark, Charles Matthews, Giftlite, MathKnight, Icairns, Natalya, R.e.b., YurikBot, Matikkapoika~enwiki, SmackBot, Jtwdog, CRGreathouse, Tewapack, Thijs!bot, RobHar, Vanish2, Ambrose H. Field, Hesam7, Jrdodge, SilvonenBot, Addbot, Roentgenium111, Guy1890, Citation bot, Omnipaedista, Citation bot 1, DrilBot, Mathsgg, Orenburg1, 777sms, Dylan Moreland, Helpful Pixie Bot, Teddyktchan and Anonymous: 9 • Hilbert symbol Source: https://en.wikipedia.org/wiki/Hilbert_symbol?oldid=651100997 Contributors: Charles Matthews, Giftlite, GroTsen, Owenjonesuk, Oleg Alexandrov, Salix alba, R.e.b., Cethegus, SmackBot, Matthuxtable, Madmath789, Mon4, Thijs!bot, Vanish2, Jakob.scholbach, STBot, Ixionid, Leyo, Virginia-American, Addbot, Corindo~enwiki, Citation bot, Citation bot 1, Trappist the monk, Deltahedron, Spectral sequence and Anonymous: 6 • Hilbert’s ninth problem Source: https://en.wikipedia.org/wiki/Hilbert’{}s_ninth_problem?oldid=564107151 Contributors: Charles Matthews, Jitse Niesen, Icairns, Lejean2000, GregorB, NatusRoma, R.e.b., Arthur Rubin, Mattbuck, Thijs!bot, Arcfrk, Addbot, RedBot, Gamewizard71, RjwilmsiBot, Ripchip Bot, Deltahedron, Mathematician757 and Anonymous: 1
244.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
731
• Hilbert’s twelfth problem Source: https://en.wikipedia.org/wiki/Hilbert’{}s_twelfth_problem?oldid=664108026 Contributors: Zundark, Michael Hardy, TakuyaMurata, Charles Matthews, Peruvianllama, Icairns, Jimius, Bender235, Wood Thrush, Keenan Pepper, Pgimeno~enwiki, Oleg Alexandrov, GregorB, NatusRoma, R.e.b., Mathbot, SmackBot, CRGreathouse, Myasuda, Mattbuck, Ntsimp, Singularity, Arcfrk, MystBot, Addbot, Lightbot, Citation bot, Raulshc, Gamewizard71, Carrotkit, Helpful Pixie Bot, BG19bot, Enyokoyama, Spectral sequence, Teddyktchan, Wurtemberg and Anonymous: 6 • Hurwitz problem Source: https://en.wikipedia.org/wiki/Hurwitz_problem?oldid=665963376 Contributors: Michael Hardy, Giftlite, Joel B. Lewis and Deltahedron • Hyper-finite field Source: https://en.wikipedia.org/wiki/Hyper-finite_field?oldid=646678817 Contributors: Michael Hardy, Rjwilmsi, R.e.b., Deltahedron, K9re11 and Monkbot • Hyperreal number Source: https://en.wikipedia.org/wiki/Hyperreal_number?oldid=656355272 Contributors: AxelBoldt, Mav, Bryan Derksen, Zundark, The Anome, Karen Johnson, Miguel~enwiki, Stevertigo, Michael Hardy, Kwertii, Kku, Bcrowell, Chinju, TakuyaMurata, Stevenj, BigFatBuddha, Charles Matthews, Dysprosia, Jitse Niesen, Cjmnyc, Hao2lian, Thegeekmeister, Robbot, Henrygb, Rorro, Fuelbottle, Tobias Bergemann, Giftlite, Ian Maxwell, Gene Ward Smith, Ævar Arnfjörð Bjarmason, Lethe, Wwoods, Dratman, Mellum, Rick Block, Macrakis, Fuzzy Logic, CSTAR, Eep², Qutezuce, Paul August, Pt, Rgdboer, KronicDeth, AshtonBenson, Alansohn, Keenan Pepper, Oleg Alexandrov, Linas, Graham87, BD2412, Mike Segal, FlaBot, Mathbot, Fresheneesz, Algebraist, YurikBot, Rick Norwood, Trovatore, Tetracube, Arthur Rubin, Bobguy7, SmackBot, Michaelliv, Eskimbot, “alyosha”, Meni Rosenfeld, Henning Makholm, Harryboyles, Yanwen, CRGreathouse, CmdrObot, Laplacian, Harej bot, Ken Gallager, MarkusQ, Gregbard, MC10, Gfonsecabr, Eric Lengyel, Z10x, Thenub314, Jakob.scholbach, David Eppstein, R'n'B, Hippasus, Quantling, Alan R. Fisher, Willow1729, AlnoktaBOT, Anonymous Dissident, AlleborgoBot, Katzmik, Soler97, DFRussia, Tlepp, Hans Adler, Versus22, Perchy22, DumZiBoT, Burningview, SilvonenBot, Addbot, AndersBot, Lightbot, OlEnglish, Luckas-bot, Yobot, Fraggle81, AnomieBOT, Mann jess, Citation bot, ArthurBot, Titi2~enwiki, Gio97, VladimirReshetnikov, Auclairde, Citation bot 1, Tkuvho, Fly by Night, EleferenBot, ZéroBot, Chharvey, Wmayner, Alexander E Ross, 21troyz, Paolo Lipparini, Nosuchforever, CitationCleanerBot, MikeHaskel, Stausifr, CsDix and Anonymous: 69 • Ideal class group Source: https://en.wikipedia.org/wiki/Ideal_class_group?oldid=655790874 Contributors: AxelBoldt, Zundark, Michael Hardy, Wshun, Alodyne, TakuyaMurata, Dyss, Charles Matthews, Timwi, Robbot, Georg Muntingh, Tobias Bergemann, Giftlite, Gene Ward Smith, PoolGuy, Waltpohl, Pmanderson, Vivacissamamente, Gauge, Oleg Alexandrov, Rjwilmsi, Mathbot, Chobot, Dmharvey, Grubber, Eskimbot, Nbarth, Danpovey, CRGreathouse, Mon4, Thijs!bot, Headbomb, RobHar, Ilion2, Monkus2k, Smjwilson, Hesam7, Andreas Carter, DeaconJohnFairfax, Jsondow, Virginia-American, Addbot, Roentgenium111, Luckas-bot, Yobot, DemocraticLuntz, Eric Rowland, FrescoBot, BG19bot, Deltahedron, Enyokoyama, Progressiveforest and Anonymous: 17 • Ideal norm Source: https://en.wikipedia.org/wiki/Ideal_norm?oldid=660051152 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, Pigsonthewing, SmackBot, Artakserkso, Rschwieb, GromXXVII, DavidCBryant, Omerks, JackSchmidt, Hatsoff, Addbot, Frobitz, Raulshc, Wcherowi and Anonymous: 6 • Integer Source: https://en.wikipedia.org/wiki/Integer?oldid=671475050 Contributors: AxelBoldt, Brion VIBBER, Bryan Derksen, Zundark, Andre Engels, Youssefsan, XJaM, Arvindn, Christian List, Dwheeler, Stevertigo, Michael Hardy, TakuyaMurata, Eric119, Ellywa, Ahoerstemeier, Darkwind, Salsa Shark, Ciphergoth, Nikai, Andres, Panoramix, Rob Hooft, Charles Matthews, Dysprosia, Jake Nelson, Hyacinth, Elwoz, Robbot, Moriori, Fredrik, Chris 73, Altenmann, Lowellian, Henrygb, Rholton, Jfire, OmegaMan, Hippietrail, Fuelbottle, Jleedev, Tobias Bergemann, Giftlite, Dbenbenn, Christopher Parham, Pretzelpaws, Lupin, Markus Kuhn, Arnejohs, Bovlb, Alanl, Leonard Vertighel, Knutux, Antandrus, Oneiros, Gauss, Elroch, Quota, Brianjd, Discospinster, Rich Farmbrough, Guanabot, Vsmith, Paul August, El C, Kwamikagami, Bendono, Bobo192, Dreish, .:Ajvol:., Sasquatch, Jojit fb, Deryck Chan, Obradovic Goran, Daniel Arteaga~enwiki, Jumbuck, Msh210, Alansohn, Arthena, Olegalexandrov, Wtmitchell, Velella, RJFJR, VoluntarySlave, Oleg Alexandrov, Saeed, Linas, Georgia guy, Camw, Splintax, MattGiuca, Kadri~enwiki, MONGO, Pufferfish101, PhilippWeissenbacher, Graham87, Chun-hian, Mendaliv, Jshadias, Josh Parris, Sjö, TheRingess, Salix alba, Mike Segal, Alejo2083, FlaBot, Mathbot, Jrtayloriv, Tardis, King of Hearts, Chobot, NSR, Roboto de Ajvol, YurikBot, Sceptre, Michael Slone, KSmrq, Stephenb, Gaius Cornelius, Wimt, NawlinWiki, Trovatore, Darkmeerkat, Wknight94, Analoguedragon, Saric, Theda, Arthur Rubin, Pb30, GrinBot~enwiki, Finell, A13ean, Amalthea, SmackBot, Sticky Parkin, Moeron, Incnis Mrsi, InverseHypercube, KnowledgeOfSelf, Unyoyega, AndyZ, Edgar181, Gilliam, Skizzik, Bluebot, Optikos, Jprg1966, Raja Hussain, Miquonranger03, Akanemoto, DHN-bot~enwiki, Darth Panda, Can't sleep, clown will eat me, Timothy Clemans, Vegard, Grover cleveland, Aldaron, Cybercobra, Nakon, Jiddisch~enwiki, Dreadstar, Tanyakh, DMacks, Dkusic~enwiki, BrianH123, Ace ETP, SashatoBot, Lambiam, Kuru, FrozenMan, Bjankuloski06en~enwiki, Jim.belk, IronGargoyle, 16@r, Vanished user 8ij3r8jwefi, Stwalkerster, Mets501, Quaeler, Iridescent, Igoldste, Cls14, Blehfu, Az1568, Jh12, Spindled, InvisibleK, Cxw, Wafulz, WeggeBot, SuperMidget, Kanags, Julian Mendez, Odie5533, Xantharius, JodyB, Thijs!bot, Epbr123, Koeplinger, Kahriman~enwiki, N5iln, Marek69, Kathovo, Escarbot, Mentifisto, AntiVandalBot, Seaphoto, Fnerchei, Jj137, Dryke, Karadimos, JAnDbot, GromXXVII, Davexia, Hut 8.5, Dricherby, Bongwarrior, VoABot II, JamesBWatson, ALostIguana, David Eppstein, Lukeelms, DerHexer, Esanchez7587, Seba5618, Gwern, Ksero, MartinBot, Kawehi 65, J.delanoy, Ttwo, Extransit, Guardian72, McSly, DJ1AM, NewEnglandYankee, SJP, Policron, DavidCBryant, Ja 62, Lights, X!, 28bytes, VolkovBot, Pleasantville, AlnoktaBOT, Philip Trueman, DoorsAjar, TXiKiBoT, Moogwrench, Drake Redcrest, Qxz, Imasleepviking, Metatron’s Cube, Corvus cornix, Digby Tantrum, JhsBot, Fnenu, Wolfrock, Synthebot, Seresin, AlleborgoBot, Tvinh, Katzmik, Omerks, Deconstructhis, Demmy100, SieBot, Dirtylittlesecerts, Gerakibot, RJaguar3, Triwbe, Keilana, PookeyMaster, Flyer22, Radon210, JSpung, Sbowers3, Oxymoron83, Faradayplank, JackSchmidt, Macy, OKBot, Denisarona, Sasha Callahan, Troy 07, Explicit, Blockofwood, SLSB, Loren.wilton, Elassint, ClueBot, Rumping, PipepBot, Snigbrook, The Thing That Should Not Be, GreekHouse, Jan1nad, Gaia Octavia Agrippa, P0mbal, Boing! said Zebedee, CounterVandalismBot, Harland1, LizardJr8, Liempt, ChandlerMapBot, Gakusha, DragonBot, -Midorihana-, Robbie098, Sillychiva593, Sun Creator, ParisianBlade, Sin Harvest, Computer97, La Pianista, Aitias, SoxBot III, Apparition11, BarretB, AgnosticPreachersKid, Marc van Leeuwen, ZooFari, Airplaneman, Gggh, Wyatt915, Addbot, Amyx231, Jncraton, Fieldday-sunday, Davidw1985, Leszek Jańczuk, Fluffernutter, Download, LaaknorBot, CarsracBot, 5 albert square, Numbo3-bot, Ehrenkater, Zorrobot, Legobot, Luckas-bot, DB.Gerry, The Flying Spaghetti Monster, AnomieBOT, Hairhorn, Jim1138, 9258fahsflkh917fas, Kingpin13, Jarmiz, Materialscientist, Xqbot, Capricorn42, Johnferrer, Jeffrey Mall, Jsharpminor, Almabot, Zarcillo, Ivan Shmakov, Shadowjams, Hakunamenta, Aaron Kauppi, Arammozuob, Wikipe-tan, GT1345, Citation bot 1, Maggyero, MacMed, Pinethicket, I dream of horses, 10metreh, Riitoken, Jumpytoo, AltinBotak, Merlion444, Jauhienij, Jonkerz, Isaac909, 777sms, Clader2, Tbhotch, Hornlitz, Mean as custard, TomdFr, Shafaet, TomT0m, LayZeeDK, DASHBot, EmausBot, Gfoley4, Notinlist, RenamedUser01302013, MartinThoma, Solarra, Scgtrp, Wikipelli, K6ka, TeleComNasSprVen, Fayimora, Mb5576, Lucas Thoms, ZéroBot, Fæ, Bollyjeff, Duperman01, KuduIO, Quondum, D.Lazard, Tolly4bolly, L Kensington, Donner60, Chewings72, ChuispastonBot, DASHBotAV, Anita5192, ClueBot NG, Wcherowi, Ypnypn, MelbourneStar, Rtucker913, Frietjes, Cntras, O.Koslowski, Lolimakethingsfun, Widr, Stapler9124, Theopolisme, Helpful Pixie Bot, BG19bot, Lann123, Dan653, Mark Arsten, Rm1271, Bishopgraeme, Jrobbinz1, Sweetnessman, Celceus, Mathenaire, DarafshBot, Ducknish, Lugia2453,
732
CHAPTER 244. ZAHLBERICHT
Frosty, Soda drinker, CsDix, Redd Foxx 1991, Khwartz, Dtw45, DavidLeighEllis, Galactic Citizen 299495038858569, JustinJustinJustinJustin, Mynameisrichard, Someone not using his real name, Sam Sailor, AddWittyNameHere, Jamesh1998, Catsweeds, BethNaught, Calvin.stone99, Rasheed49, This is a fake user name, UnixDaemon, This is another fake user name, GeoffreyT2000, Eduvgugvuv, Samvandervlies, Jj 1213 wiki, Esquivalience, Glitterstars23, Hgccbjhfsh, Kkk2000, SteakMuncher, Phuasien, KasparBot, Kafishabbir and Anonymous: 675 • Isomorphism extension theorem Source: https://en.wikipedia.org/wiki/Isomorphism_extension_theorem?oldid=621667856 Contributors: Zundark, Michael Hardy, Giftlite, AtticusRyan, Cronholm144, Alaibot, Vanish2, ImPerfectHacker, Geometry guy, Niceguyedc, Brad7777, Qetuth and Anonymous: 3 • Iwasawa theory Source: https://en.wikipedia.org/wiki/Iwasawa_theory?oldid=670922703 Contributors: Zundark, TakuyaMurata, Nanshu, Salsa Shark, Charles Matthews, Giftlite, PoolGuy, Waltpohl, Vivacissamamente, Bender235, Gauge, Rje, Oleg Alexandrov, Japanese Searobin, Rjwilmsi, R.e.b., Holomorph, Dmharvey, Gareth Jones, Eskimbot, Makyen, Flamelai, CRGreathouse, Xtv, Headbomb, Nnn9245, RobHar, MartinBot, Kesal, VolkovBot, JohnBlackburne, Hesam7, Minded17, AlleborgoBot, Jrdodge, WikHead, Addbot, DOI bot, Ringspectrum, FrescoBot, Lagelspeil, Citation bot 1, Trappist the monk, ZéroBot, Suslindisambiguator, Brad7777, Boodlepounce, Justincheng12345bot, Deltahedron, Spectral sequence, Melonkelon, Wikid77b, KasparBot and Anonymous: 23 • Jacobson–Bourbaki theorem Source: https://en.wikipedia.org/wiki/Jacobson%E2%80%93Bourbaki_theorem?oldid=626907610 Contributors: Giftlite, Rjwilmsi, R.e.b., Niceguyedc, Yobot, Trappist the monk, Mark viking and Anonymous: 1 • Krasner’s lemma Source: https://en.wikipedia.org/wiki/Krasner’{}s_lemma?oldid=631037583 Contributors: Michael Hardy, Rjwilmsi, RobHar, Arch dude, FrescoBot, RjwilmsiBot, Math.zchen, Deltahedron, Spectral sequence, Davidbrink, K9re11 and Anonymous: 3 • Kronecker–Weber theorem Source: https://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theorem?oldid=635704950 Contributors: Michael Hardy, Charles Matthews, Psychonaut, Giftlite, Gene Ward Smith, Bender235, Mdd, Eric Kvaalen, Rjwilmsi, R.e.b., Korg, YurikBot, SmackBot, BeteNoir, Mon4, RobHar, Dugwiki, David Eppstein, R'n'B, Geometry guy, Nickhann, Alexbot, Ncsinger, WikHead, Addbot, Luckas-bot, Yobot, Kilom691, Citation bot 1, Trappist the monk, ZéroBot, Quandle, Brad7777, Enyokoyama, Jmstahl and Anonymous: 9 • Kummer theory Source: https://en.wikipedia.org/wiki/Kummer_theory?oldid=644949604 Contributors: Zundark, Michael Hardy, Charles Matthews, Dysprosia, Fredrik, MathMartin, UtherSRG, Giftlite, Bensaccount, Eequor, Vivacissamamente, Cacycle, Gauge, EmilJ, Oleg Alexandrov, R.e.b., Marco Streng, YurikBot, Dmharvey, Dbfirs, Zhw, Lhf, CRGreathouse, RobHar, Vanish2, Hesam7, B2smith, Addbot, LaaknorBot, Yobot, BenzolBot, Brad7777, Enyokoyama, GreenKeeper17 and Anonymous: 21 • Lafforgue’s theorem Source: https://en.wikipedia.org/wiki/Lafforgue’{}s_theorem?oldid=635261140 Contributors: Michael Hardy, Giftlite, R.e.b., Headbomb, Slawekb, D.Lazard, Brad7777, K9re11 and Anonymous: 3 • Langlands dual group Source: https://en.wikipedia.org/wiki/Langlands_dual_group?oldid=638730958 Contributors: Charles Matthews, Giftlite, R.e.b., CmdrObot, Headbomb, RobHar, Changbao, Mild Bill Hiccup, Julian Birdbath, Jackmcbarn and Anonymous: 5 • Langlands–Deligne local constant Source: https://en.wikipedia.org/wiki/Langlands%E2%80%93Deligne_local_constant?oldid=639752535 Contributors: TakuyaMurata, R.e.b., Makyen, Headbomb, Yobot and Anonymous: 1 • Lazard’s universal ring Source: https://en.wikipedia.org/wiki/Lazard’{}s_universal_ring?oldid=614968696 Contributors: Michael Hardy, R.e.b., Headbomb, Yobot, D.Lazard, Mark viking and Anonymous: 1 • Leopoldt’s conjecture Source: https://en.wikipedia.org/wiki/Leopoldt’{}s_conjecture?oldid=656506268 Contributors: Michael Hardy, Charles Matthews, Giftlite, Bender235, R.e.b., SmackBot, JoshuaZ, Jim.belk, CBM, Headbomb, Nnn9245, RobHar, Fabrictramp, David Eppstein, VolkovBot, XLinkBot, MatthewVanitas, Addbot, Roentgenium111, Koppas, Omnipaedista, Suslindisambiguator, Helpful Pixie Bot, Spectral sequence and Anonymous: 7 • Levi-Civita field Source: https://en.wikipedia.org/wiki/Levi-Civita_field?oldid=635358815 Contributors: Michael Hardy, BRG, Nikitadanilov, Gro-Tsen, Arthur Rubin, Loadmaster, Fashionslide, Cheesefondue, ZooFari, Yobot, Joule36e5, Erik9bot, Tkuvho, Per Ardua, ClueBot NG, Gergely.Szekely, CsDix and Anonymous: 8 • Linked field Source: https://en.wikipedia.org/wiki/Linked_field?oldid=606061329 Contributors: David Eppstein, Ironholds, RjwilmsiBot, Deltahedron and Spectral sequence • Liouville’s theorem (differential algebra) Source: https://en.wikipedia.org/wiki/Liouville’{}s_theorem_(differential_algebra)?oldid= 643848894 Contributors: Michael Hardy, A5, Gandalf61, Rjwilmsi, [email protected], Zunaid, SmackBot, Headbomb, MystBot, Addbot, Berberisb, CaroleHenson, Brad7777, Chodu Das and Anonymous: 5 • List of algebraic number theory topics Source: https://en.wikipedia.org/wiki/List_of_algebraic_number_theory_topics?oldid=657215788 Contributors: Michael Hardy, Charles Matthews, Shizhao, ZeroOne, Oleg Alexandrov, Mathbot, Fram, Fplay, Syrcatbot, Valoem, RobHar, Danger, Robert Skyhawk, Addbot, RTG, Verbal, Yobot, Brad7777, 7WOLFGANG SCHWARZ7 and Anonymous: 2 • List of number fields with class number one Source: https://en.wikipedia.org/wiki/List_of_number_fields_with_class_number_one? oldid=638674332 Contributors: Michael Hardy, Giftlite, Rjwilmsi, SmackBot, Sljaxon, JoshuaZ, RobHar, Marc van Leeuwen, Roentgenium111, Uncia, Yobot, Erik9bot, Gkopp, BG19bot, Deltahedron and Anonymous: 5 • Local class field theory Source: https://en.wikipedia.org/wiki/Local_class_field_theory?oldid=666366801 Contributors: Zundark, TakuyaMurata, Charles Matthews, Giftlite, Vivacissamamente, R.e.b., Wavelength, Dmharvey, SmackBot, CBM, Headbomb, RobHar, Dugwiki, Vanish2, JohnBlackburne, Kyle the bot, Hesam7, Recipr, Addbot, Ringspectrum, Chrangers, BG19bot, Tomoliver163, Brad7777, Deltahedron, Brirush, GravRidr and Anonymous: 7 • Local Euler characteristic formula Source: https://en.wikipedia.org/wiki/Local_Euler_characteristic_formula?oldid=634820303 Contributors: Michael Hardy, SmackBot, Headbomb, RobHar, David Eppstein, VQuakr, Addbot, Neonorange, AnomieBOT, Mr. Guye and K9re11 • Local field Source: https://en.wikipedia.org/wiki/Local_field?oldid=613287739 Contributors: AxelBoldt, Michael Hardy, Alodyne, TakuyaMurata, Ciphergoth, Loren Rosen, Charles Matthews, Greenrd, Michael Larsen, SEWilco, Georg Muntingh, Giftlite, Fropuff, Cvalente, Vivacissamamente, Mlm42, Oleg Alexandrov, Wavelength, Dmharvey, RussBot, SEWilcoBot, Arthur Rubin, SmackBot, Bluebot, MalafayaBot, Ligulembot, Lambiam, Headbomb, RobHar, Magioladitis, Vanish2, Policron, TXiKiBoT, Plclark, YonaBot, JackSchmidt, CharlesGillingham, Addbot, Yobot, Citation bot, Overlocal, Ringspectrum, FrescoBot, Fontainejm, John of Reading, Noodle snacks, Learner344, Chrangers, Helpful Pixie Bot, Adele362, BG19bot, Tomoliver163 and Anonymous: 15
244.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
733
• Local Fields Source: https://en.wikipedia.org/wiki/Local_Fields?oldid=657216628 Contributors: Michael Hardy, TakuyaMurata, Nabla, R.e.b., Valoem, Danrok, Headbomb, Xeno, JaGa, Henrytucker, PV=nRT, Ringspectrum, Trappist the monk, CobraBot, Helpful Pixie Bot, Gorobay, Deltahedron and Anonymous: 3 • Local Langlands conjectures Source: https://en.wikipedia.org/wiki/Local_Langlands_conjectures?oldid=643822257 Contributors: TakuyaMurata, Charles Matthews, Rjwilmsi, R.e.b., Headbomb, RobHar, David Eppstein, Kilom691, Jonesey95, Helpful Pixie Bot, Deltahedron, K9re11 and Anonymous: 2 • Lubin–Tate formal group law Source: https://en.wikipedia.org/wiki/Lubin%E2%80%93Tate_formal_group_law?oldid=669778901 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, Gro-Tsen, Mdd, Rjwilmsi, R.e.b., Chef aka Pangloss, Headbomb, Yobot, FrescoBot, Frietjes, Tomoliver163, Shotaro Makisumi, Deltahedron, Mark viking and Anonymous: 3 • Lüroth’s theorem Source: https://en.wikipedia.org/wiki/L%C3%BCroth’{}s_theorem?oldid=638874336 Contributors: BDD, Eubot, David Eppstein, MenoBot, D.Lazard, Maimonid, K9re11, Monkbot and Anonymous: 3 • Maillet’s determinant Source: https://en.wikipedia.org/wiki/Maillet’{}s_determinant?oldid=650888118 Contributors: Michael Hardy, Rjwilmsi, R.e.b., Leyo, Trappist the monk, Mark viking and K9re11 • Minimal polynomial (field theory) Source: https://en.wikipedia.org/wiki/Minimal_polynomial_(field_theory)?oldid=607165604 Contributors: TakuyaMurata, Jitse Niesen, Fredrik, Macrakis, Shreevatsa, Eubot, Michael Slone, Bh3u4m, JoergenB, Cwkmail, Jtyard, Addbot, Jarble, Yobot, Ptbotgourou, AnomieBOT, Talkloud, MondalorBot, Encryptionscheme, EmausBot, WikitanvirBot, Honestrosewater, Wcherowi, Helpful Pixie Bot, Mheasley, Maimonid, Monkbot and Anonymous: 11 • Minkowski space (number field) Source: https://en.wikipedia.org/wiki/Minkowski_space_(number_field)?oldid=604300753 Contributors: Ironholds, Deltahedron and Anonymous: 1 • Mode of a linear field Source: https://en.wikipedia.org/wiki/Mode_of_a_linear_field?oldid=646598991 Contributors: Michael Hardy, Waldir, Kri, JMO, Gogo Dodo, Katharineamy, Eeekster, Yobot, AnomieBOT, Hhhippo, DoctorKubla, Reatlas, Ekramul hossain, Hoppadoodledo, CORPORATE SOLUTIONS WORLD AHEAD and Anonymous: 2 • Modulus (algebraic number theory) Source: https://en.wikipedia.org/wiki/Modulus_(algebraic_number_theory)?oldid=657216410 Contributors: Michael Hardy, WhisperToMe, R.e.b., SmackBot, Valoem, Headbomb, RobHar, Vanish2, JackSchmidt, RjwilmsiBot, The Stick Man, Matthiaspaul, Helpful Pixie Bot, Deltahedron, Tango303 and Anonymous: 5 • Monogenic field Source: https://en.wikipedia.org/wiki/Monogenic_field?oldid=620455502 Contributors: Giftlite, MarnetteD, Loadmaster, CRGreathouse, CmdrObot, Vanish2, Marksmith55, CaroleHenson, Helpful Pixie Bot, Spectral sequence, Mark viking and Anonymous: 3 • Multiplicative group Source: https://en.wikipedia.org/wiki/Multiplicative_group?oldid=670283852 Contributors: Zundark, Charles Matthews, Giftlite, Rgdboer, Mdd, SDC, Mathbot, Laurentius, Arthur Rubin, Stepa, Dan Hoey, Noleander, CmdrObot, Jakob.scholbach, Minimiscience, Daniele.tampieri, JackSchmidt, He7d3r, MystBot, Addbot, Yobot, AnomieBOT, Erik9bot, RjwilmsiBot, Quondum, D.Lazard, Brad7777, Jeremy112233, StriatumPDM, CsDix and Anonymous: 7 • Nagata’s conjecture Source: https://en.wikipedia.org/wiki/Nagata’{}s_conjecture?oldid=631072726 Contributors: Michael Hardy, R.e.b., LokiClock, Trappist the monk, K9re11, Stivenson and Anonymous: 1 • Narrow class group Source: https://en.wikipedia.org/wiki/Narrow_class_group?oldid=597640786 Contributors: Charles Matthews, Dmharvey, KSmrq, CRGreathouse, RobHar, Gwern, Roentgenium111, BG19bot and Anonymous: 1 • Newton polygon Source: https://en.wikipedia.org/wiki/Newton_polygon?oldid=670017584 Contributors: Michael Hardy, Charles Matthews, Diberri, Giftlite, Jason Quinn, Rich Farmbrough, Firsfron, Aghitza, Valoem, SpecZ, Exceptg, David Eppstein, Myrizio, STBot, Zermalo, JackSchmidt, Qtpi0113, Addbot, Uscitizenjason, Trappist the monk, Flegmon, Quandle, Vinícius Machado Vogt, ChrisGualtieri, Arcandam, Makecat-bot, Citizentoad, Jochen Burghardt, K9re11, WordSeventeen and Anonymous: 9 • Non-abelian class field theory Source: https://en.wikipedia.org/wiki/Non-abelian_class_field_theory?oldid=600804536 Contributors: Charles Matthews, Salix alba, Vanish2, Enyokoyama and Anonymous: 1 • Norm form Source: https://en.wikipedia.org/wiki/Norm_form?oldid=645970979 Contributors: Charles Matthews, SmackBot, David Eppstein and Erik9bot • Norm group Source: https://en.wikipedia.org/wiki/Norm_group?oldid=643542171 Contributors: TakuyaMurata, Malcolma, AvicBot, Qetuth and K9re11 • Normal basis Source: https://en.wikipedia.org/wiki/Normal_basis?oldid=672065974 Contributors: Zundark, Michael Hardy, Charles Matthews, Gene Ward Smith, CryptoDerk, Rich Farmbrough, Gauge, Rjwilmsi, SmackBot, Xaosflux, David Eppstein, TreasuryTag, Addbot, Constructive editor, Louperibot, Helpful Pixie Bot, Deltahedron, Lichfielder and Anonymous: 3 • Normal extension Source: https://en.wikipedia.org/wiki/Normal_extension?oldid=660061556 Contributors: AxelBoldt, Zundark, Michael Hardy, MathKnight, WalterM, Art LaPella, EmilJ, Oleg Alexandrov, Bgwhite, Michael Slone, Russell C. Sibley, Maksim-e~enwiki, JCSantos, Gutworth, Khazar, Timmie.merc, Thijs!bot, Konradek, RobHar, Jakob.scholbach, LokiClock, Kyle the bot, Plclark, COBot, Ideal gas equation, Alexbot, Bender2k14, Sandrobt, Addbot, Expz, Luckas-bot, Yobot, FrescoBot, TobeBot, Helpful Pixie Bot and Anonymous: 11 • P-adic Hodge theory Source: https://en.wikipedia.org/wiki/P-adic_Hodge_theory?oldid=606732043 Contributors: TakuyaMurata, Rjwilmsi, R.e.b., Headbomb, RobHar, Yobot, BTotaro, Helpful Pixie Bot, Brad7777, Dkmiller271 and Anonymous: 1 • P-adic number Source: https://en.wikipedia.org/wiki/P-adic_number?oldid=672721871 Contributors: Damian Yerrick, AxelBoldt, Mav, Bryan Derksen, Zundark, The Anome, Toby Bartels, PierreAbbat, Miguel~enwiki, Patrick, Chas zzz brown, Coopercc, Michael Hardy, Dominus, Chinju, TakuyaMurata, Minesweeper, Looxix~enwiki, Ciphergoth, AugPi, Rotem Dan, Iseeaboar, Ideyal, Revolver, A5, Charles Matthews, Dcoetzee, Dysprosia, Populus, SirJective, Robbot, Gandalf61, MathMartin, Aetheling, Fuelbottle, Tobias Bergemann, Tosha, Giftlite, Gene Ward Smith, Lethe, Fropuff, Dratman, Waltpohl, Jason Quinn, Eequor, John Palkovic, Chowbok, Keith Edkins, CryptoDerk, Dnas, TheBlueWizard, Mikolt, TedPavlic, Pjacobi, Chad okere, H00kwurm, ReiVaX, Paul August, Bender235, Ben Standeven, Gauge, Bluap, Stephen Bain, Jumbuck, Eric Kvaalen, Sligocki, Wadems, Kusma, Oleg Alexandrov, Joriki, Linas, MFH, Tachyon², Isnow, Graham87, RxS, MarSch, R.e.b., Marozols, FlaBot, Mathbot, Maxal, Chobot, Bgwhite, YurikBot, RobotE, Ilanpi,
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Taejo, KSmrq, Trovatore, Długosz, Crasshopper, Arthur Rubin, Kier07, JeffBurdges, Singingwolfboy, Brentt, SmackBot, Adam majewski, Incnis Mrsi, Melchoir, Eskimbot, Chris the speller, Oli Filth, Jallotta, Nbarth, Emurphy42, Daqu, Qpt, Lambiam, Khazar, Loadmaster, Rill2503456, Thatcher, Jbolden1517, Jafet, CRGreathouse, Mon4, Dharma6662000, Thijs!bot, Julian Brown, MetsBot, David Eppstein, Hairchrm, Ttwo, Insp nf, Haziel, VolkovBot, LokiClock, Anonymous Dissident, PaulTanenbaum, AlleborgoBot, SieBot, BotMultichill, JackSchmidt, Paul Clapham, Amahoney, Classicalecon, DeaconJohnFairfax, DFRussia, Alexbot, Hans Adler, Addbot, Roentgenium111, Eric Drexler, DOI bot, Download, Kisbesbot, AHbot, Jarble, Yobot, MinorProphet, AnomieBOT, Nippashish, FrescoBot, Citation bot 4, Dark Charles, Heptadecagon, ElNuevoEinstein, Mjs1991, Numericana, E.V.Krishnamurthy, Jowa fan, Drusus 0, Codygunton, Quondum, Frietjes, Helpful Pixie Bot, Snorri, BG19bot, Dmcallas, Chmarkine, Dexbot, Deltahedron, Mark viking, Jodosma, PetrusJohannes, BethNaught, GeoffreyT2000 and Anonymous: 108 • P-adic order Source: https://en.wikipedia.org/wiki/P-adic_order?oldid=637987748 Contributors: Zundark, Tobias Bergemann, Giftlite, Gene Ward Smith, Peruvianllama, Chad okere, MFH, Maxal, Incnis Mrsi, Melchoir, CRGreathouse, Vanish2, Addbot, Matěj Grabovský, Eric Rowland, Deathbeast, Super-real dance, Larsborn, Cdvf, K9re11, Monkbot and Anonymous: 4 • P-adically closed field Source: https://en.wikipedia.org/wiki/P-adically_closed_field?oldid=574335274 Contributors: Zundark, GroTsen, Nikkimaria, Uncia, Erik9bot, Citation bot 1, CitationCleanerBot, Mark viking and Anonymous: 1 • Parshin chain Source: https://en.wikipedia.org/wiki/Parshin_chain?oldid=650834946 Contributors: Rjwilmsi, R.e.b., Trappist the monk, Deltahedron, Mark viking and K9re11 • Perfect field Source: https://en.wikipedia.org/wiki/Perfect_field?oldid=587177721 Contributors: AxelBoldt, Zundark, TakuyaMurata, Charles Matthews, EmilJ, CBM, Myasuda, Konradek, Headbomb, RobHar, Albmont, Policron, Addbot, Roentgenium111, Legobot, Yobot, Ptbotgourou, Darij, Brained, Quondum, Helpful Pixie Bot, CitationCleanerBot, Danneks and Anonymous: 7 • Perfectoid Source: https://en.wikipedia.org/wiki/Perfectoid?oldid=648789182 Contributors: TakuyaMurata, MushroomCloud, Trlovejoy, Derek R Bullamore, PamD, David Eppstein, Dthomsen8 and Deltahedron • Polynomial basis Source: https://en.wikipedia.org/wiki/Polynomial_basis?oldid=644686745 Contributors: Mm, Charles Matthews, CryptoDerk, ArnoldReinhold, Michael Slone, Afitzy, WinBot, Dimawik, Niceguyedc, Yobot, AnomieBOT, Citation bot 1, D.Lazard and Anonymous: 5 • Power residue symbol Source: https://en.wikipedia.org/wiki/Power_residue_symbol?oldid=672343704 Contributors: Pigsonthewing, Virginia-American, Yobot, LilHelpa, Omnipaedista, BudgieJane, Spectral sequence and Anonymous: 3 • Primary extension Source: https://en.wikipedia.org/wiki/Primary_extension?oldid=522518033 Contributors: TakuyaMurata, SmackBot, Bte99, Qetuth and Deltahedron • Primitive element (finite field) Source: https://en.wikipedia.org/wiki/Primitive_element_(finite_field)?oldid=672064770 Contributors: Michael Hardy, Maxal, MvH, Arun ks, Thijs!bot, RobHar, Vanish2, David Eppstein, Addbot, Yobot, Nargaque, D.Lazard, Wcherowi, MerlIwBot, Qetuth, Enyokoyama, Monkbot and Anonymous: 8 • Primitive element theorem Source: https://en.wikipedia.org/wiki/Primitive_element_theorem?oldid=614415276 Contributors: Zundark, William Avery, Michael Hardy, Oyd11, Charles Matthews, MathMartin, Giftlite, Gene Ward Smith, Billymac00, Dfeldmann, Simetrical, Bgwhite, Algebraist, Bo Jacoby, Tac-Tics, Asmeurer, Vanish2, Austinmohr, Geometry guy, Niceguyedc, ATC2, Sandrobt, Addbot, Yobot, TaBOT-zerem, Point-set topologist, IhorLviv, MerlIwBot, Brad7777, SiriusLH, Darvii and Anonymous: 17 • Primitive polynomial (field theory) Source: https://en.wikipedia.org/wiki/Primitive_polynomial_(field_theory)?oldid=669945029 Contributors: Zundark, Michael Hardy, Oyd11, Charles Matthews, Dcoetzee, Giftlite, CryptoDerk, Calair, 3mta3, Oleg Alexandrov, Maxal, Roboto de Ajvol, RussBot, Mmernex, Jjbeard~enwiki, Lhf, Thijs!bot, RobHar, WinBot, GiM, Asmeurer, Vanish2, JamesBWatson, Jozwiakjohn, R00723r0, UffeHThygesen, KirbenS, JackSchmidt, Wdwd, Marc van Leeuwen, Addbot, Cxz111, AnomieBOT, AKappa, Quondum, D.Lazard, Wcherowi, Quanfui, BattyBot, Jacob Jakobsen, Prbs man and Anonymous: 21 • Principalization (algebra) Source: https://en.wikipedia.org/wiki/Principalization_(algebra)?oldid=647212430 Contributors: Michael Hardy, Rjwilmsi, Ksoileau, TutterMouse, Yobot, AnomieBOT, DanielConstantinMayer, Iamthecheese44, Deltahedron, WMartin74, GreenKeeper17, Elysion and Monkbot • Profinite integer Source: https://en.wikipedia.org/wiki/Profinite_integer?oldid=659545750 Contributors: Michael Hardy, TakuyaMurata and K9re11 • Proofs of Fermat’s theorem on sums of two squares Source: https://en.wikipedia.org/wiki/Proofs_of_Fermat’{}s_theorem_on_sums_ of_two_squares?oldid=665835293 Contributors: Michael Hardy, Chinju, Charles Matthews, Dcoetzee, Giftlite, D6, Rich Farmbrough, Zenohockey, Woohookitty, Karam.Anthony.K, Rjwilmsi, Salix alba, Magidin, Mathbot, Algebraist, Zvika, Psiphiorg, JCSantos, Colonies Chris, Daqu, Madmath789, LDH, CRGreathouse, Phauly, Limweizhong, Jaerik, Thom Tyrrell, JeffTowers, Strategist333, Arcfrk, GirasoleDE, NowhereDense, Marc van Leeuwen, Laudan08, AnomieBOT, Full-date unlinking bot, HighCrossRuff, Wisapi, Sapphorain, Tawarama, BG19bot, Chmarkine, ChrisGualtieri, Teddyktchan and Anonymous: 27 • Proofs of quadratic reciprocity Source: https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity?oldid=609838872 Contributors: Charles Matthews, Lowellian, C S, Woohookitty, Salix alba, Holomorph, Dmharvey, RussBot, KSmrq, Zvika, -Ozone-, Digana, Jim.belk, CRGreathouse, ShelfSkewed, Stebulus, RobHar, David Eppstein, Chridd, Marc van Leeuwen, Virginia-American, Anticipation of a New Lover’s Arrival, The, Helpful Pixie Bot, BG19bot and Anonymous: 10 • Pseudo algebraically closed field Source: https://en.wikipedia.org/wiki/Pseudo_algebraically_closed_field?oldid=585862571 Contributors: Zundark, TakuyaMurata, Charles Matthews, EmilJ, R.e.b., Algebraist, YurikBot, Matikkapoika~enwiki, Barylior, Addbot, Fred Gandt, Qetuth, ChrisGualtieri, Deltahedron, Spectral sequence, Lichfielder and Anonymous: 3 • Pseudo-finite field Source: https://en.wikipedia.org/wiki/Pseudo-finite_field?oldid=635281918 Contributors: Michael Hardy, TakuyaMurata, Rjwilmsi, R.e.b., Zujua, Deltahedron, Mark viking and Monkbot • Purely inseparable extension Source: https://en.wikipedia.org/wiki/Purely_inseparable_extension?oldid=627025559 Contributors: Michael Hardy, TakuyaMurata, MathMartin, Rjwilmsi, R.e.b., Bgwhite, Niceguyedc, AnomieBOT and Trappist the monk • Pythagoras number Source: https://en.wikipedia.org/wiki/Pythagoras_number?oldid=632659917 Contributors: Michael Hardy, David Eppstein, Duoduoduo, Wcherowi, Deltahedron and Anonymous: 1 • Pythagorean field Source: https://en.wikipedia.org/wiki/Pythagorean_field?oldid=607161665 Contributors: Michael Hardy, R.e.b., Vaughan Pratt, Headbomb, David Eppstein, Yobot, Gongshow, AnomieBOT, Tkuvho, RjwilmsiBot, Deltahedron, Spectral sequence and Anonymous: 2
244.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
735
• Quadratic field Source: https://en.wikipedia.org/wiki/Quadratic_field?oldid=669129787 Contributors: Michael Hardy, TakuyaMurata, Revolver, Charles Matthews, MathMartin, Giftlite, Gene Ward Smith, Lupin, Jason Quinn, Icairns, Vivacissamamente, Oleg Alexandrov, Linas, R.e.b., YurikBot, Wavelength, Nbarth, Jim.belk, Valoem, CRGreathouse, Mon4, Xantharius, RobHar, Vanish2, VolkovBot, Hesam7, Arcfrk, Gerakibot, Alexbot, Uschilch, Addbot, Luckas-bot, Yobot, AnomieBOT, Raulshc, Ringspectrum, John85, TobeBot, Mathtyke, Quondum, D.Lazard, MerlIwBot, Enyokoyama, Spectral sequence and Anonymous: 11 • Quadratic integer Source: https://en.wikipedia.org/wiki/Quadratic_integer?oldid=667634470 Contributors: Zundark, Dominus, Chinju, TakuyaMurata, R.e.b., Bgwhite, DavidHouse~enwiki, SmackBot, Incnis Mrsi, BlackFingolfin, AbsolutDan, CRGreathouse, Gyopi, RobHar, TomyDuby, Dessources, Selinger, GirasoleDE, JackSchmidt, Frobitz, Yobot, AnomieBOT, Anne Bauval, Raulshc, Tonyxty, D.Lazard, Acrazydiamond, Gergely.Szekely, CitationCleanerBot, Brad7777, and Anonymous: 7 • Quadratic reciprocity Source: https://en.wikipedia.org/wiki/Quadratic_reciprocity?oldid=669816059 Contributors: AxelBoldt, Michael Hardy, Modster, TakuyaMurata, GTBacchus, Notheruser, Revolver, Charles Matthews, Dysprosia, .mau., Robbot, Fredrik, Lowellian, Ojigiri~enwiki, Giftlite, Harp, Herbee, Wmahan, Gubbubu, Profvk, Andyabides, Tsemii, Quota, Vivacissamamente, Bender235, El C, EmilJ, Owenjonesuk, 3mta3, Tavdy79, Eric Kvaalen, Smithereens~enwiki, WojciechSwiderski~enwiki, Marudubshinki, Salix alba, Mike Segal, Sodin, Chobot, YurikBot, Dmharvey, RussBot, Lenthe, Gaius Cornelius, Commeca, DYLAN LENNON~enwiki, Hv, Ronyclau, GrinBot~enwiki, SmackBot, Rouenpucelle, Chris the speller, Gutworth, DHN-bot~enwiki, Ohconfucius, Lambiam, JoshuaZ, Alethiophile, Drae, LDH, CRGreathouse, Wafulz, ShelfSkewed, Achangeisasgoodasa, Sobreira, LeoDaVinci, Arch dude, PhilKnight, Vanish2, Johnbibby, David Eppstein, Ttwo, VAFisher, Nwbeeson, Policron, Nxavar, Geometry guy, GirasoleDE, SieBot, Chridd, ClueBot, Justin W Smith, Marc van Leeuwen, Virginia-American, Addbot, Download, דוד55, Yobot, Xqbot, GrouchoBot, Lefschetz, Citation bot 1, Hansomkan, Full-date unlinking bot, Miracle Pen, John of Reading, WikitanvirBot, KHamsun, Slawekb, Javanaut, Helpful Pixie Bot, Nysnamovois, Kaspuhler, Pquadick, Larsborn, Deltahedron, Jsinick, Jianhui67, Teddyktchan and Anonymous: 73 • Quadratically closed field Source: https://en.wikipedia.org/wiki/Quadratically_closed_field?oldid=549751119 Contributors: Michael Hardy, Shay Falador, EmausBot and Deltahedron • Quartic reciprocity Source: https://en.wikipedia.org/wiki/Quartic_reciprocity?oldid=617724570 Contributors: Giftlite, Scott Ritchie, Rjwilmsi, Maxal, Sodin, RussBot, Grafen, Chris the speller, Mgiganteus1, Jim.belk, ElommolE, CRGreathouse, ShelfSkewed, Fconaway, Rumpuscat, Virginia-American, Addbot, Yobot, Lefschetz, Citation bot 1, CrowzRSA, RjwilmsiBot, Jowa fan, Helpful Pixie Bot, Deltahedron, K9re11 and Anonymous: 4 • Quasi-algebraically closed field Source: https://en.wikipedia.org/wiki/Quasi-algebraically_closed_field?oldid=652803927 Contributors: Michael Hardy, Charles Matthews, Giftlite, Rjwilmsi, R.e.b., SmackBot, Martarius, Addbot, David Brink, Omnipaedista, Ringspectrum, ZéroBot, D.Lazard, Uni.Liu, BG19bot, Deltahedron, Spectral sequence, SoSivr and Anonymous: 5 • Quasi-finite field Source: https://en.wikipedia.org/wiki/Quasi-finite_field?oldid=522664942 Contributors: Zundark, Paul August, Oleg Alexandrov, Graham87, R.e.b., Dmharvey, SmackBot, Headbomb, RobHar, Nono64, Citation bot, Zfeinst, Helpful Pixie Bot, Deltahedron and Anonymous: 1 • Quaternionic structure Source: https://en.wikipedia.org/wiki/Quaternionic_structure?oldid=616912083 Contributors: David Eppstein, YohanN7, Ironholds, Kiefer.Wolfowitz, Deltahedron and Spectral sequence • Ramification (mathematics) Source: https://en.wikipedia.org/wiki/Ramification_(mathematics)?oldid=660655129 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, Zigger, Peruvianllama, Superborsuk, Vivacissamamente, Lemontea, Circeus, Oleg Alexandrov, Daira Hopwood, R.e.b., Dmharvey, RussBot, Gwaihir, Zwobot, Matikkapoika~enwiki, Titopao, Nbarth, Valoem, Jafet, CBM, Thijs!bot, RobHar, AntiVandalBot, Vanish2, Jakob.scholbach, MartinBot, Boston, STBotD, TXiKiBoT, Hesam7, GirasoleDE, Addbot, Ringspectrum, ElNuevoEinstein, Xanchester, Enyokoyama, Spectral sequence, OlofBergvall, Bjcnet and Anonymous: 24 • Ramification group Source: https://en.wikipedia.org/wiki/Ramification_group?oldid=663711890 Contributors: Michael Hardy, TakuyaMurata, Charles Matthews, Malcolma, Crasshopper, Valoem, Mack2, Scott Tillinghast, Houston TX, Jakob.scholbach, Download, Yobot, FrescoBot, AvicBot, Helpful Pixie Bot, Qetuth, Deltahedron, Spectral sequence and Anonymous: 5 • Ramification theory of valuations Source: https://en.wikipedia.org/wiki/Ramification_theory_of_valuations?oldid=566208190 Contributors: TakuyaMurata, Gbrading, RobHar, Qetuth, Spectral sequence, Mark viking and Anonymous: 2 • Rational number Source: https://en.wikipedia.org/wiki/Rational_number?oldid=671930454 Contributors: AxelBoldt, Brion VIBBER, Bryan Derksen, Zundark, Tarquin, Jan Hidders, Andre Engels, XJaM, Christian List, Toby Bartels, PierreAbbat, Roadrunner, FvdP, Stevertigo, Patrick, Michael Hardy, Wshun, MartinHarper, Ixfd64, TakuyaMurata, Mdebets, Ciphergoth, AugPi, Andres, Evercat, Panoramix, Pizza Puzzle, Hashar, Hawthorn, Revolver, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Greenrd, Hyacinth, Thue, McKay, Guppy, Denelson83, Robbot, Romanm, Mayooranathan, Thunderbolt16, Henrygb, Borislav, Fuelbottle, Lupo, Tobias Bergemann, Giftlite, Gene Ward Smith, Ævar Arnfjörð Bjarmason, Lethe, Dissident, Fropuff, Dratman, Guanaco, Bovlb, Jason Quinn, Jorge Stolfi, Nayuki, Tagishsimon, Rheun, Ato, Antandrus, MarkSweep, Bob.v.R, Vina, Scott Burley, Ehamberg, Lostchicken, Mormegil, Discospinster, Notinasnaid, Paul August, El C, EmilJ, Deanos, Bobo192, Elipongo, Jung dalglish, Blotwell, Deryck Chan, Obradovic Goran, Jumbuck, Msh210, Alansohn, Ncik~enwiki, Silver86, Wtmitchell, Velella, L33th4x0rguy, Mikeo, Btornado, Oleg Alexandrov, Linas, StradivariusTV, Prashanthns, Graham87, BD2412, SixWingedSeraph, Yurik, Zzedar, Jshadias, Josh Parris, Sdornan, Salix alba, The wub, Bhadani, Yamamoto Ichiro, David H Braun (1964), DVdm, YurikBot, Wavelength, Stephenb, Pseudomonas, NawlinWiki, Rick Norwood, E rulez, Hennobrandsma, Charlie Wiederhold, Lt-wiki-bot, Gesslein, GrinBot~enwiki, Crystallina, Hydrogen Iodide, Zerida, Unyoyega, Yamaguchi , Gilliam, Skizzik, Persian Poet Gal, Raymond arritt, Raja Hussain, MalafayaBot, Akanemoto, DHN-bot~enwiki, Can't sleep, clown will eat me, Shunpiker, Grover cleveland, Daqu, Nakon, Dreadstar, NickPenguin, Salamurai, Bidabadi~enwiki, Vinaiwbot~enwiki, Ck lostsword, SashatoBot, Lambiam, Nishkid64, Btritchie, Kuru, CorvetteZ51, Cronholm144, Gobonobo, Jim.belk, Ekrub-ntyh, Loadmaster, Dr Greg, Mets501, Lee Carre, Quaeler, Jazzcello, Majora4, ILikeThings, JForget, CRGreathouse, Wafulz, Penbat, SuperMidget, Cydebot, Worthingtonse, Boardhead, Epbr123, Koeplinger, Martin Hogbin, Marek69, Wmasterj, AbcXyz, Escarbot, Ju66l3r, AntiVandalBot, Vvidetta, Edokter, Braindrain0000, JAnDbot, MER-C, Smiddle, .anacondabot, Connormah, Bongwarrior, VoABot II, Twsx, WODUP, Avicennasis, Seberle, JoergenB, DerHexer, GermanX, Aschmitz, Tokidoki27, MartinBot, Kostisl, Jarhed, J.delanoy, Katalaveno, Stwitzel, SJP, Policron, CompuChip, Angular, DavidCBryant, Wikieditor06, VolkovBot, Johan1298~enwiki, TallNapoleon, AlnoktaBOT, VasilievVV, TXiKiBoT, Maximillion Pegasus, Dendodge, Broadbot, Hrundi Bakshi, Maxim, Wolfrock, Synthebot, Allan1114, Monty845, AlleborgoBot, LuigiManiac, EmxBot, Omerks, Demmy, Bfpage, SieBot, Legion fi, Yintan, Bentogoa, Flyer22, Tiptoety, Macy, OKBot, Diego Grez, Angielaj, Randomblue, Dolphin51, Troy 07, Joe Photon, ClueBot, Rumping, PipepBot, Fyyer, The Thing That Should Not Be, Cliff, Jpcs, Dylan620, Paritybit, Jusdafax, Eeekster, Cenarium, Jotterbot, Aitias, Katanada, SoxBot III, Crazy Boris with a red beard, AgnosticPreachersKid, Spitfire, Crapme, Zrs 12, NellieBly, Dnvrfantj, RyanCross, HexaChord, Addbot, Proofreader77, ConCompS, Friginator, Ronhjones, TutterMouse, Fieldday-sunday, Skyezx, LaaknorBot,
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CHAPTER 244. ZAHLBERICHT
LinkFA-Bot, Jaydec, AgadaUrbanit, Numbo3-bot, Tide rolls, MZaplotnik, Teles, LuK3, Luckas-bot, Yobot, Tempodivalse, DemocraticLuntz, 9258fahsflkh917fas, Kingpin13, Materialscientist, Erikekahn, OllieFury, Xelnx, ArthurBot, Xqbot, TinucherianBot II, Capricorn42, Doctor Rosenberg, Isheden, Hackabhihack, RibotBOT, The Wiki Octopus, Aaron Kauppi, Stlrams22, Lothar von Richthofen, DivineAlpha, Tkuvho, Pinethicket, ShadowRangerRIT, I dream of horses, Adlerbot, MarcelB612, BigDwiki, Jujutacular, Reconsider the static, Tim1357, ItsZippy, Jonkerz, Dinamik-bot, Vrenator, JuanGabrielRobalino, Stroppolo, Luhar1997, TjBot, Bento00, TomT0m, EmausBot, Khalidmathematics, Slightsmile, TuHan-Bot, Wikipelli, ReySquared, Alpha Quadrant, Quondum, Mburdis, Wayne Slam, L Kensington, Donner60, Chewings72, Puffin, Orange Suede Sofa, 28bot, ClueBot NG, This lousy T-shirt, MikuMikuCookie, Dfarrell07, Mpaa, Ichliebepferde, Spel-Punc-Gram, Kasirbot, Widr, WikiPuppies, Calabe1992, Kelsi1122, Sheilds, Darksonn, Ihatechickens214, Mercrutio, Nick white03, Glacialfox, Bodema, Kishugoyal, Teammm, Pratyya Ghosh, Dexbot, FoCuSandLeArN, Mcash001, Scharan09, Lugia2453, Epicgenius, Encyclopedia 12, Harry styles5555554, Imperial Marshmallow, AmaryllisGardener, Maxtheaxe1999, Ray Lightyear, Zenibus, JDiala, Wellset, Wikiwonka7777, Moorelife, Programmer 112, Wilson Widyadhana, Anirudh Babu, ChamithN, DemonKiller3527, Engilukol albert and Anonymous: 596 • Rational reciprocity law Source: https://en.wikipedia.org/wiki/Rational_reciprocity_law?oldid=627077865 Contributors: Michael Hardy, Giftlite, Rjwilmsi, R.e.b., JL-Bot, LilHelpa, Trappist the monk and Deltahedron • Rational variety Source: https://en.wikipedia.org/wiki/Rational_variety?oldid=648137367 Contributors: Michael Hardy, Gabbe, Charles Matthews, Robbot, Giftlite, Gene Ward Smith, Waltpohl, Dave donaghy, Joe Decker, R.e.b., Jerome Kelly, Jdaloner, Niceguyedc, Addbot, EhsanQ, Lightbot, Yobot, Kilom691, Citation bot 1, DrilBot, Trappist the monk, ZéroBot, D.Lazard, Uni.Liu, Jerry science, Enyokoyama, Maimonid, Rkinser, Monkbot and Anonymous: 3 • Ray class field Source: https://en.wikipedia.org/wiki/Ray_class_field?oldid=653254279 Contributors: BD2412, R.e.b., Cesium 133, Niceguyedc, BG19bot and Anonymous: 2 • Real closed field Source: https://en.wikipedia.org/wiki/Real_closed_field?oldid=670551371 Contributors: Michael Hardy, Chinju, Charles Matthews, Jitse Niesen, Greenrd, David.Monniaux, Aleph4, Giftlite, Gene Ward Smith, Lethe, Icairns, Sam Hocevar, Sctfn, D6, TedPavlic, Paul August, Zaslav, EmilJ, Nortexoid, Oleg Alexandrov, Imaginatorium, Linas, Nahabedere, Chenxlee, R.e.b., John Baez, Mathbot, Algebraist, Danlaycock, SmackBot, Incnis Mrsi, Chris the speller, MalafayaBot, Voceditenore, Mets501, CRGreathouse, WinBot, Jakob.scholbach, JaGa, Barraki, TXiKiBoT, Nono le petit robot~enwiki, Niceguyedc, Hans Adler, Addbot, Legobot, Luckas-bot, AnomieBOT, Tkuvho, Jujutacular, Marcus0107, SporkBot, Jcimpric, Erick GR, Helpful Pixie Bot, BG19bot, Deltahedron, Spectral sequence, AHusain314, Jochen Burghardt and Anonymous: 16 • Reciprocity law Source: https://en.wikipedia.org/wiki/Reciprocity_law?oldid=627077913 Contributors: Michael Hardy, TakuyaMurata, Notheruser, Charles Matthews, Giftlite, Johnflux, Klemen Kocjancic, BD2412, R.e.b., Xyzzyplugh, JL-Bot, Virginia-American, Yobot, Boleyn3, Trappist the monk, BG19bot, Deltahedron, Spectral sequence and Anonymous: 5 • Regular extension Source: https://en.wikipedia.org/wiki/Regular_extension?oldid=653686675 Contributors: Zundark, TakuyaMurata, Malcolma, SmackBot, CmdrObot, David Callan, Bte99, Qetuth, Deltahedron and Anonymous: 1 • Regular prime Source: https://en.wikipedia.org/wiki/Regular_prime?oldid=663536829 Contributors: XJaM, Schneelocke, Charles Matthews, David Shay, Phil Boswell, Fredrik, MathMartin, Henrygb, PrimeFan, Giftlite, Herbee, Dratman, Peter Kwok, Bender235, Eric Kvaalen, Pouya, Rjwilmsi, R.e.b., Mathbot, Maxal, Gwaihir, Darker Dreams, JHCaufield, Reyk, PrimeHunter, Throwawayhack, Cybercobra, DRLB, Iffy, CRGreathouse, Boemanneke, PamD, Headbomb, RobHar, Vanish2, David Eppstein, STBotD, VolkovBot, LokiClock, Wikiisawesome, Le Pied-bot~enwiki, Niceguyedc, Safek, Jsondow, MystBot, Addbot, Lightbot, Balabiot, Luckas-bot, Yobot, Ptbotgourou, Cristiano Toàn, LilHelpa, TobeBot, RjwilmsiBot, EmausBot, Jurvetson2, Toshio Yamaguchi, Sapphorain, Tesla91, Mikhail Ryazanov, Supermint, Jam000qaz, Helpful Pixie Bot, Spectral sequence, Noyster and Anonymous: 27 • Resolvent (Galois theory) Source: https://en.wikipedia.org/wiki/Resolvent_(Galois_theory)?oldid=666689642 Contributors: Michael Hardy, Giftlite, Lockley, Maxim Leyenson, Jack Daw, David Eppstein, Bender2k14, YouRang?, D.Lazard, BG19bot, J58660, Tristanhands, Tentinator and Anonymous: 5 • Rigid analytic space Source: https://en.wikipedia.org/wiki/Rigid_analytic_space?oldid=627078175 Contributors: Michael Hardy, TakuyaMurata, Giftlite, R.e.b., SmackBot, S.H.C., CmdrObot, CBM, Trappist the monk, GAGARIG and Anonymous: 4 • Ring class field Source: https://en.wikipedia.org/wiki/Ring_class_field?oldid=637443011 Contributors: R.e.b., TexasAndroid, Sarahj2107, David Eppstein, Yobot, AnomieBOT, Jesse V., Mark viking and Anonymous: 1 • Ring of integers Source: https://en.wikipedia.org/wiki/Ring_of_integers?oldid=654367375 Contributors: AxelBoldt, Zundark, Michael Hardy, TakuyaMurata, Charles Matthews, Malcohol, Giftlite, Fropuff, Jason Quinn, Eric Kvaalen, Simetrical, Salix alba, [email protected], Chrispounds, Dan131m, SmackBot, Incnis Mrsi, Bluebot, SMP, Vaughan Pratt, CRGreathouse, Mon4, RobHar, Vanish2, Selinger, Botx, VolkovBot, Ferengi, Omerks, Addbot, Roentgenium111, Luckas-bot, FancyMouse, Rckrone, John85, EmausBot, Deltahedron, Citizentoad, Purnendu Karmakar, K9re11, R6i6N4Hin and Anonymous: 11 • Ring of p-adic periods Source: https://en.wikipedia.org/wiki/Ring_of_p-adic_periods?oldid=564756977 Contributors: Michael Hardy, Bgwhite, Yobot, Dkmiller271 and Anonymous: 1 • Rupture field Source: https://en.wikipedia.org/wiki/Rupture_field?oldid=606790265 Contributors: Gandalf61, Rich Farmbrough, SmackBot, Alaibot, Albmont, SurJector, Kernel Saunters, Addbot, Luckas-bot, Asfarer, Erik9bot, Jujutacular, Helpful Pixie Bot, Brad7777, Deltahedron and Anonymous: 5 • S-unit Source: https://en.wikipedia.org/wiki/S-unit?oldid=646570598 Contributors: Edward, Woohookitty, RobHar, Vanish2, David Eppstein, RjwilmsiBot, Frietjes, Helpful Pixie Bot, Deltahedron and Anonymous: 4 • Separable extension Source: https://en.wikipedia.org/wiki/Separable_extension?oldid=659808099 Contributors: AxelBoldt, Zundark, Edward, TakuyaMurata, Charles Matthews, Dysprosia, Vivacissamamente, Shotwell, Mazi, Bender235, Vipul, EmilJ, Keenan Pepper, Oleg Alexandrov, R.e.b., YurikBot, SmackBot, Eskimbot, Mets501, WLior, Tac-Tics, Dragonflare82, Thijs!bot, RobHar, Jakob.scholbach, Etale, Allispaul, LokiClock, Wedhorn, Ideal gas equation, Niceguyedc, Alexbot, Bender2k14, Addbot, Luckas-bot, FredrikMeyer, AnomieBOT, Citation bot, LilHelpa, GrouchoBot, Point-set topologist, Hkhk59333, Citation bot 1, John of Reading, Codygunton, D.Lazard, BendelacBOT, Deltahedron, Mark viking and Anonymous: 23 • Separable polynomial Source: https://en.wikipedia.org/wiki/Separable_polynomial?oldid=671423651 Contributors: Edward, Michael Hardy, Charles Matthews, Giftlite, EmilJ, Oleg Alexandrov, Linas, HannsEwald, Laurentius, Michael Slone, [email protected], MvH, Konradek, RobHar, Vanish2, David Eppstein, VolkovBot, Kchu, Niceguyedc, Marc van Leeuwen, Addbot, Yinweichen, Honestrosewater, D.Lazard, PhnomPencil and Anonymous: 6
244.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
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• Serre’s conjecture II (algebra) Source: https://en.wikipedia.org/wiki/Serre’{}s_conjecture_II_(algebra)?oldid=622499266 Contributors: Michael Hardy, Bender235, Rjwilmsi, RobHar, Exceptg, MuffledThud, RobinK, Spectral sequence and Anonymous: 1 • Simple extension Source: https://en.wikipedia.org/wiki/Simple_extension?oldid=595508851 Contributors: TakuyaMurata, MathMartin, Giftlite, Marudubshinki, Salix alba, YurikBot, Dmharvey, Michael Slone, Eskimbot, Petr Matas, Vanish2, Hesam7, JackSchmidt, Ideal gas equation, NuclearWarfare, Sandrobt, MystBot, Addbot, TobeBot, D.Lazard, JordiGH, Wcherowi, Deltahedron, Darvii and Anonymous: 5 • Skolem–Mahler–Lech theorem Source: https://en.wikipedia.org/wiki/Skolem%E2%80%93Mahler%E2%80%93Lech_theorem?oldid= 661961785 Contributors: Michael Hardy, Giftlite, Rjwilmsi, Rschwieb, David Eppstein, Kope, Addbot, Yobot, LilHelpa, Sławomir Biały, Bezik and Brad7777 • Splitting field Source: https://en.wikipedia.org/wiki/Splitting_field?oldid=653598342 Contributors: Michael Hardy, Looxix~enwiki, Charles Matthews, Giftlite, EmilJ, LutzL, Burn, Oleg Alexandrov, Mandarax, Dmharvey, Incnis Mrsi, Eskimbot, Jprg1966, MvH, TooMuchMath, Harej bot, JAnDbot, GromXXVII, .anacondabot, Magioladitis, TomyDuby, Policron, LokiClock, Codairem, SurJector, He7d3r, Bender2k14, Marc van Leeuwen, Addbot, Dyaa, PV=nRT, Yobot, TaBOT-zerem, AnomieBOT, Cpryby, Jujutacular, Greenfernglade, Pgdoyle, EmausBot, Fly by Night, Razghandi, Nosuchforever, Lundril, Nigellwh and Anonymous: 27 • Splitting of prime ideals in Galois extensions Source: https://en.wikipedia.org/wiki/Splitting_of_prime_ideals_in_Galois_extensions? oldid=660767998 Contributors: Edward, Charles Matthews, Peruvianllama, Rich Farmbrough, EmilJ, Dmharvey, SmackBot, Kilo-Lima, CRGreathouse, RobHar, Frettloe, Jakob.scholbach, Lejarrag, XLinkBot, Addbot, Armin Straub, Yobot, Raulshc, Aliotra, Cenkner, EdwardH, Jmcommelin, Enyokoyama, Bjcnet, Progressiveforest and Anonymous: 19 • Square class Source: https://en.wikipedia.org/wiki/Square_class?oldid=593666769 Contributors: Bearcat, Malcolma, David Eppstein, AnomieBOT, Skeesix, BattyBot, DoctorKubla and CouchSurfer222 • Stark conjectures Source: https://en.wikipedia.org/wiki/Stark_conjectures?oldid=660246366 Contributors: Michael Hardy, Charles Matthews, Giftlite, Eric Shalov, Bender235, R.e.b., Gutworth, CBM, Forgetfulfunctor, David Eppstein, Arcfrk, B2smith, DumZiBoT, Lightbot, Trappist the monk, Deltahedron, Enyokoyama and Anonymous: 2 • Strassmann’s theorem Source: https://en.wikipedia.org/wiki/Strassmann’{}s_theorem?oldid=636933224 Contributors: Charles Matthews, Giftlite, R.e.b., Sodin, Myasuda, Ntsimp, Sullivan.t.j, DavidCBryant, Yobot, Trappist the monk, Qetuth and Brirush • Stufe (algebra) Source: https://en.wikipedia.org/wiki/Stufe_(algebra)?oldid=666853993 Contributors: Michael Hardy, Joriki, Algebraist, CRGreathouse, Yobot, AnomieBOT, ViolaPlayer, Chutznik, Deltahedron, Spectral sequence and Anonymous: 3 • Superreal number Source: https://en.wikipedia.org/wiki/Superreal_number?oldid=636362364 Contributors: Michael Hardy, Gene Ward Smith, CyborgTosser, Ben Standeven, Helopticor, Keenan Pepper, Oleg Alexandrov, Linas, Lockley, R.e.b., Nomet, SmackBot, Zero sharp, Vorpal22, Hippasus, AlnoktaBOT, Reidonius, Phe-bot, CheepnisAroma, DFRussia, Addbot, Luckas-bot, Yobot, Tkuvho, Trappist the monk, ZéroBot, NHCLS and Anonymous: 6 • Supersingular prime (for an elliptic curve) Source: https://en.wikipedia.org/wiki/Supersingular_prime_(for_an_elliptic_curve)?oldid= 659153066 Contributors: Michael Hardy, Lowellian, Giftlite, Bender235, Gutworth, Cardano~enwiki, Addbot, Roentgenium111, SlavaPhD, Helpful Pixie Bot, Deltahedron and Anonymous: 3 • Symbol (number theory) Source: https://en.wikipedia.org/wiki/Symbol_(number_theory)?oldid=647811128 Contributors: Michael Hardy, Giftlite, R.e.b. and K9re11 • Takagi existence theorem Source: https://en.wikipedia.org/wiki/Takagi_existence_theorem?oldid=657215562 Contributors: Edward, Charles Matthews, Giftlite, Gene Ward Smith, R.e.b., Mathbot, Sodin, SmackBot, BeteNoir, Valoem, Biblbroks, Dugwiki, WhatamIdoing, R'n'B, Ambrose H. Field, Kyle the bot, Addbot, Yobot, Ringspectrum, Cobaltcigs, Brad7777, Enyokoyama and Anonymous: 3 • Tate cohomology group Source: https://en.wikipedia.org/wiki/Tate_cohomology_group?oldid=621607794 Contributors: Rjwilmsi, R.e.b., TeaDrinker, SmackBot, RobHar, David Eppstein, LokiClock, Ks0stm, AnomieBOT, Quotient group, Mqtrinh, K9re11 and Anonymous: 5 • Tate duality Source: https://en.wikipedia.org/wiki/Tate_duality?oldid=631036451 Contributors: R.e.b., Yobot, Trappist the monk, K9re11 and Anonymous: 1 • Tate’s thesis Source: https://en.wikipedia.org/wiki/Tate’{}s_thesis?oldid=659933885 Contributors: Charles Matthews, Giftlite, R.e.b., Gutworth, Myasuda, Headbomb, David Eppstein, Roentgenium111, RjwilmsiBot, Alonamit, Symmetries134, Oddprimes and Anonymous: 3 • Teichmüller character Source: https://en.wikipedia.org/wiki/Teichm%C3%BCller_character?oldid=648782839 Contributors: Michael Hardy, Charles Matthews, BD2412, R.e.b., Headbomb, RobHar, LokiClock, Yobot, AnomieBOT, Mark viking, K9re11 and Anonymous: 2 • Teichmüller cocycle Source: https://en.wikipedia.org/wiki/Teichm%C3%BCller_cocycle?oldid=653938284 Contributors: Rjwilmsi, R.e.b., Yobot and K9re11 • Tensor product of fields Source: https://en.wikipedia.org/wiki/Tensor_product_of_fields?oldid=657213881 Contributors: AxelBoldt, Zundark, Michael Hardy, TakuyaMurata, Charles Matthews, Waltpohl, Vivacissamamente, Gauge, Oleg Alexandrov, Woohookitty, R.e.b., Peter Grey, Gwaihir, SmackBot, Jushi, Nbarth, LkNsngth, Allansteel, Zeteg, Valoem, CRGreathouse, Ksoileau, Thijs!bot, Headbomb, RobHar, Vanish2, Error792, Laurusnobilis, BBrucker2, Addbot, LilHelpa, MauritsBot, Ringspectrum, Quondum and Anonymous: 7 • Thin set (Serre) Source: https://en.wikipedia.org/wiki/Thin_set_(Serre)?oldid=628758643 Contributors: Michael Hardy, Charles Matthews, Firsfron, RussBot, SmackBot, Politepunk, Barylior, Niceguyedc, Addbot, Luckas-bot, Ringspectrum, D.Lazard, Ultra snozbarg, Deltahedron and Anonymous: 2 • Timeline of class field theory Source: https://en.wikipedia.org/wiki/Timeline_of_class_field_theory?oldid=644480515 Contributors: TakuyaMurata, R.e.b., JL-Bot and K9re11 • Totally imaginary number field Source: https://en.wikipedia.org/wiki/Totally_imaginary_number_field?oldid=643182667 Contributors: RobHar and John of Reading • Totally real number field Source: https://en.wikipedia.org/wiki/Totally_real_number_field?oldid=634561940 Contributors: Vicki Rosenzweig, Charles Matthews, Waltpohl, Drbreznjev, Grammarbot, Chenxlee, R.e.b., Lenthe, Thijs!bot, RobHar, Hesam7, Librarian2, Alexbot, Addbot, Roentgenium111, Erik9bot, Bineapple and Anonymous: 1
738
CHAPTER 244. ZAHLBERICHT
• Tower of fields Source: https://en.wikipedia.org/wiki/Tower_of_fields?oldid=632934103 Contributors: Pol098, Crasshopper, RobHar, Quondum, Technopop.tattoo, K9re11 and Anonymous: 1 • Transcendence degree Source: https://en.wikipedia.org/wiki/Transcendence_degree?oldid=621668228 Contributors: AxelBoldt, Zundark, Michael Hardy, Poor Yorick, Charles Matthews, Michael Larsen, Giftlite, EmilJ, Oleg Alexandrov, Chenxlee, Benandorsqueaks, Bo Jacoby, SmackBot, Vina-iwbot~enwiki, David Eppstein, JackSchmidt, Niceguyedc, Virginia-American, MystBot, Addbot, Luckas-bot, Yobot, GeometryGirl, Artem M. Pelenitsyn, Solomonfromfinland and Anonymous: 13 • Tschirnhaus transformation Source: https://en.wikipedia.org/wiki/Tschirnhaus_transformation?oldid=650002350 Contributors: Michael Hardy, Charles Matthews, Dino, Eyu100, Lunch, Shelog~enwiki, Lambiam, Jasperdoomen, Mr. Granger, Addbot, Luckas-bot, Yobot, Halothane, Wikielwikingo, D.Lazard, Qetuth and Anonymous: 3 • Tsen rank Source: https://en.wikipedia.org/wiki/Tsen_rank?oldid=561189047 Contributors: Michael Hardy, Excirial, AvicBot, Accedie, Spectral sequence and Anonymous: 1 • Twisted polynomial ring Source: https://en.wikipedia.org/wiki/Twisted_polynomial_ring?oldid=649383909 Contributors: JustAGal, Ahecht, LokiClock, AnomieBOT, Deltahedron and Dirk Basson • U-invariant Source: https://en.wikipedia.org/wiki/U-invariant?oldid=646572977 Contributors: Michael Hardy, David Eppstein, Frietjes, Deltahedron and Ultramarineflycatcher • Unique factorization domain Source: https://en.wikipedia.org/wiki/Unique_factorization_domain?oldid=672652228 Contributors: AxelBoldt, Zundark, Michael Hardy, Alodyne, Dominus, Chinju, TakuyaMurata, Karada, Loren Rosen, Revolver, Charles Matthews, MathMartin, Sverdrup, Tobias Bergemann, Giftlite, Waltpohl, Pmanderson, Noisy, Pt, Rgdboer, Ceyockey, Tbsmith, Cyclotronwiki, Hypercube~enwiki, Rjwilmsi, R.e.b., Chobot, Algebraist, YurikBot, [email protected], Gwaihir, Ms2ger, Evilbu, MalafayaBot, Nbarth, Justplainuncool, CRGreathouse, HenningThielemann, Mon4, RobHar, Seaphoto, Vanish2, Albmont, Jakob.scholbach, David Eppstein, Lucky Eight, Daniel5Ko, Imadada, LokiClock, Plclark, Rhubbarb, BOTarate, Marc van Leeuwen, Addbot, LaaknorBot, Luckas-bot, Yobot, Amirobot, Nallimbot, AnomieBOT, Xqbot, Depassp, Omnipaedista, Point-set topologist, Charvest, Citation bot 1, TobeBot, Trappist the monk, Math2481, E273b773, Deltahedron, Galaas, Mark viking, Cyrapas, Mathmensch, K9re11, Strecosaurus, Monkbot, Wikiixtp, GeoffreyT2000 and Anonymous: 50 • Unit (ring theory) Source: https://en.wikipedia.org/wiki/Unit_(ring_theory)?oldid=667990156 Contributors: AxelBoldt, TakuyaMurata, Revolver, Charles Matthews, Dysprosia, MathMartin, Giftlite, Smjg, Fropuff, Vivacissamamente, HasharBot~enwiki, Jérôme, Oleg Alexandrov, MFH, Salix alba, FlaBot, VKokielov, Maxal, RexNL, Gaius Cornelius, BOT-Superzerocool, Arthur Rubin, Reyk, SmackBot, Incnis Mrsi, Melchoir, SMP, Jim.belk, Alma Teao Wilson, Mlepicki, Cydebot, Keyi, Kilva, JAnDbot, Magioladitis, Peskydan, Trumpet marietta 45750, LokiClock, Omerks, Marc van Leeuwen, Pitt SATSA, Virginia-American, CàlculIntegral, Addbot, דניאל ב., Fryed-peach, TaBOT-zerem, AnomieBOT, Anne Bauval, Erik9bot, Rckrone, Ebony Jackson, Quondum, ClueBot NG, Jochen Burghardt, Zhongmou Zhang and Anonymous: 16 • Universal quadratic form Source: https://en.wikipedia.org/wiki/Universal_quadratic_form?oldid=532994389 Contributors: Ryan Reich, R.e.b. and Deltahedron • Valuation (algebra) Source: https://en.wikipedia.org/wiki/Valuation_(algebra)?oldid=669096757 Contributors: Zundark, Michael Hardy, Ciphergoth, Charles Matthews, Dysprosia, Greenrd, Onebyone, Pfortuny, Giftlite, Gene Ward Smith, Dratman, PhotoBox, Noisy, Gauge, EmilJ, Nortexoid, Lachaume, Oleg Alexandrov, Joriki, Linas, R.e.b., FlaBot, Gaius Cornelius, Melchoir, Ppntori, ERcheck, Mhss, CRGreathouse, CBM, Gregbard, Mon4, Headbomb, RobHar, Magioladitis, R'n'B, Jeepday, Daniele.tampieri, DomSemaca, LokiClock, JackSchmidt, DionysosProteus, He7d3r, Addbot, LaaknorBot, Rubinbot, NOrbeck, FrescoBot, Ebony Jackson, Boobarkee, TobeBot, EmausBot, Anita5192, MOSNUM Bot, BG19bot, Cjsh716, Spectral sequence, K9re11, Mgkrupa, GeoffreyT2000, JMP EAX, Alakzi and Anonymous: 22 • Valuation ring Source: https://en.wikipedia.org/wiki/Valuation_ring?oldid=670295900 Contributors: AxelBoldt, Zundark, Michael Hardy, TakuyaMurata, Charles Matthews, Dmitri83, Giftlite, Gene Ward Smith, Rgdboer, Guardian of Light, Rjwilmsi, Salix alba, R.e.b., Chef aka Pangloss, Nbarth, Vina-iwbot~enwiki, Mets501, Rschwieb, RobHar, Srostami, Kyle the bot, JackSchmidt, Legobot, AnomieBOT, Anne Bauval, Ringspectrum, Tkuvho, FoxBot, Trappist the monk, EmausBot, Snotbot, Gergely.Szekely, BG19bot, Elfinit, Brad7777, Spectral sequence, Tall human and Anonymous: 20 • Weil group Source: https://en.wikipedia.org/wiki/Weil_group?oldid=664819367 Contributors: TakuyaMurata, Charles Matthews, Rjwilmsi, R.e.b., WLior, Headbomb, RobHar, Jakob.scholbach, Helpful Pixie Bot, Enyokoyama, Mark viking and Anonymous: 2 • Zahlbericht Source: https://en.wikipedia.org/wiki/Zahlbericht?oldid=627092086 Contributors: Michael Hardy, Giftlite, R.e.b., JL-Bot, Kilom691 and Trappist the monk
244.5.2
Images
• File:2adic12480.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/99/2adic12480.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Melchoir • File:3-adic_integers_with_dual_colorings.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/ce/3-adic_integers_with_ dual_colorings.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Melchoir • File:3-adic_metric_on_integers.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9a/3-adic_metric_on_integers.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Incnis Mrsi • File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public domain Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs) • File:Archimedean_property.png Source: https://upload.wikimedia.org/wikipedia/commons/3/3f/Archimedean_property.png License: Public domain Contributors: ru:Файл:Аксиома Архимеда.png Original artist: Arkadius • File:Arithmetic_symbols.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a3/Arithmetic_symbols.svg License: Public domain Contributors: Own work Original artist: This vector image was created with Inkscape by Elembis, and then manually replaced. • File:ClassExtensionAndArtinTransfer.png Source: https://upload.wikimedia.org/wikipedia/en/b/b8/ClassExtensionAndArtinTransfer. png License: CC0 Contributors: ? Original artist: ?
244.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
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• File:Coclass1Tree2Groups.png Source: https://upload.wikimedia.org/wikipedia/en/0/0b/Coclass1Tree2Groups.png License: CC0 Contributors: ? Original artist: ? • File:Coclass1Tree3Groups.png Source: https://upload.wikimedia.org/wikipedia/en/2/2d/Coclass1Tree3Groups.png License: CC0 Contributors: ? Original artist: ? • File:Coclass1Tree5Groups.png Source: https://upload.wikimedia.org/wikipedia/en/2/25/Coclass1Tree5Groups.png License: CC0 Contributors: ? Original artist: ? • File:Coclass2TreeQType33.png Source: https://upload.wikimedia.org/wikipedia/en/2/2d/Coclass2TreeQType33.png License: CC0 Contributors: ? Original artist: ? • File:Coclass2TreeUType33.png Source: https://upload.wikimedia.org/wikipedia/en/2/2c/Coclass2TreeUType33.png License: CC0 Contributors: ? Original artist: ? • File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Original artist: ? • File:Cyclic_group.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/5f/Cyclic_group.svg License: CC BY-SA 3.0 Contributors: • Cyclic_group.png Original artist: • derivative work: Pbroks13 (talk) • File:DAB_list_gray.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/8c/DAB_list_gray.svg License: CC-BY-SA-3.0 Contributors: modified versions from File:Disambig gray.svg Original artist: Edokter (modified version) • File:Dedekind.jpeg Source: https://upload.wikimedia.org/wikipedia/commons/c/ca/Dedekind.jpeg License: Public domain Contributors: http://dbeveridge.web.wesleyan.edu/wescourses/2001f/chem160/01/Photo_Gallery_Science/Dedekind/FrameSet.htm Original artist: not found • File:Diagonal_argument.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/85/Diagonal_argument.svg License: CC-BYSA-3.0 Contributors: Own work Original artist: Cronholm144 • File:Diagram_of_a_Newton_Polygon_Convex_hull.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/79/Diagram_of_ a_Newton_Polygon_Convex_hull.svg License: CC BY-SA 3.0 Contributors: Gnuplot, version 4.4 script Original artist: Uscitizenjason
• File:Discriminant49CubicFieldFundamentalDomain.png Source: https://upload.wikimedia.org/wikipedia/commons/e/e5/Discriminant49CubicFieldFunda png License: CC BY-SA 3.0 Contributors: Own work Original artist: RobHar • File:Discriminant_of_cubic_polynomials..png Source: https://upload.wikimedia.org/wikipedia/commons/8/81/Discriminant_of_cubic_ polynomials..png License: CC BY-SA 4.0 Contributors: Own work Original artist: Salix alba • File:Disquisitiones-Arithmeticae-p133.jpg Source: https://upload.wikimedia.org/wikipedia/commons/8/84/Disquisitiones-Arithmeticae-p133. jpg License: Public domain Contributors: ? Original artist: ? • File:Disqvisitiones-800.jpg Source: https://upload.wikimedia.org/wikipedia/commons/e/e3/Disqvisitiones-800.jpg License: Public domain Contributors: ? Original artist: ? • File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: The Tango! Desktop Project. Original artist: The people from the Tango! project. And according to the meta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (although minimally).” • File:Eisenstein-quadratic-reciprocity-1.svg Source: https://upload.wikimedia.org/wikipedia/en/7/72/Eisenstein-quadratic-reciprocity-1. svg License: CC-BY-SA-2.5 Contributors: ? Original artist: ? • File:Eisenstein-quadratic-reciprocity-2.svg Source: https://upload.wikimedia.org/wikipedia/en/8/8e/Eisenstein-quadratic-reciprocity-2. svg License: CC-BY-SA-2.5 Contributors: ? Original artist: ? • File:Eisenstein-quadratic-reciprocity-3.svg Source: https://upload.wikimedia.org/wikipedia/en/6/62/Eisenstein-quadratic-reciprocity-3. svg License: CC-BY-SA-2.5 Contributors: ? Original artist: ? • File:Eisenstein-quadratic-reciprocity-4.svg Source: https://upload.wikimedia.org/wikipedia/en/5/5c/Eisenstein-quadratic-reciprocity-4. svg License: CC-BY-SA-2.5 Contributors: ? Original artist: ? • File:EisensteinPrimes-01.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/17/EisensteinPrimes-01.svg License: Public domain Contributors: self-made with Mathematica Original artist: Fropuff • File:EpiAndArtinTransfers.png Source: https://upload.wikimedia.org/wikipedia/en/f/f6/EpiAndArtinTransfers.png License: CC0 Contributors: ? Original artist: ? • File:EpiAndDerivedQuotients.png Source: https://upload.wikimedia.org/wikipedia/en/8/86/EpiAndDerivedQuotients.png License: CC0 Contributors: ? Original artist: ? • File:FactorThruDerivedQuotient.png Source: https://upload.wikimedia.org/wikipedia/en/5/56/FactorThruDerivedQuotient.png License: CC0 Contributors: ? Original artist: ? • File:FrequencyCoclass2Type33Sporadic.tiff Source: https://upload.wikimedia.org/wikipedia/en/7/7e/FrequencyCoclass2Type33Sporadic. tiff License: CC0 Contributors: ? Original artist: ? • File:Gnome-searchtool.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1e/Gnome-searchtool.svg License: LGPL Contributors: http://ftp.gnome.org/pub/GNOME/sources/gnome-themes-extras/0.9/gnome-themes-extras-0.9.0.tar.gz Original artist: David Vignoni • File:Golden_spiral_in_rectangles.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/91/Golden_spiral_in_rectangles.svg License: Public domain Contributors: Own work Original artist: Luiz Real • File:Latex_integers.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c1/Latex_integers.svg License: CC-BY-SA-3.0 Contributors: it’s one of LaTeX fonts, converted in SVG by myself Original artist: Alessio Damato
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• File:Lattice_diagram_of_Q_adjoin_a_cube_root_of_2_and_a_primitive_cube_root_of_1,_its_subfields,_and_Galois_groups. svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b6/Lattice_diagram_of_Q_adjoin_a_cube_root_of_2_and_a_primitive_ cube_root_of_1%2C_its_subfields%2C_and_Galois_groups.svg License: CC BY-SA 3.0 Contributors: Created in LaTeX by the following code: \documentclass[12pt]{article} \thispagestyle{empty} \usepackage{tikz} \usepackage{amsfonts} \begin{document} \begin{tikzpicture}[node distance=2cm] \node (Qw-o) {$\mathbb{Q}(\omega, \theta)$}; \node (Qo) [below of=Qw-o] {$\mathbb{Q}(\theta)$}; \node (Qwo) [right of=Qo] {$\mathbb{Q}(\omega \theta)$}; \node (Qw2o) [right of=Qwo] {$\mathbb{Q}(\omega^2 \theta)$}; \node (Qw) [below left of=Qo] {$\mathbb{Q}(\omega)$}; \node (Q) [below right of=Qw] {$\mathbb{Q}$}; \node (1ff2) [below right of=Qw2o] {$\{1, f, f^2\}$}; \node (1g) [above right of=1ff2] {$\{1, g\}$}; \node (1gf) [right of=1g] {$\{1, gf\}$}; \node (1gf2) [right of=1gf] {$\{1, gf^2\}$}; \node (G) [below right of=1ff2] {$\{1, f, f^2, g, gf, gf^2\}$}; \node (1) [above of=1g] {$\{1\}$}; \draw (Q) -- (Qo); \draw (Q) -- (Qw); \draw (Q) -- (Qwo); \draw (Q) -- (Qw2o); \draw (Qo) -- (Qwo); \draw (Qw) -- (Qw-o); \draw (Qwo) -- (Qw-o); \draw (Qw2o) -- (Qw-o); \draw (G) -- (1ff2); \draw (G) -- (1g); \draw (G) -- (1gf); \draw (G) -- (1gf2); \draw (1ff2) -- (1); \draw (1g) -- (1); \draw (1gf) -- (1); \draw (1gf2) -- (1); \end{tikzpicture} \end{document} Original artist: Self • File:Lattice_diagram_of_Q_adjoin_the_positive_square_roots_of_2_and_3,_its_subfields,_and_Galois_groups.svg Source: https: //upload.wikimedia.org/wikipedia/commons/a/ad/Lattice_diagram_of_Q_adjoin_the_positive_square_roots_of_2_and_3%2C_its_subfields% 2C_and_Galois_groups.svg License: CC BY-SA 3.0 Contributors: Created in LaTeX by the following code: \documentclass[12pt]{article} \thispagestyle{empty} \usepackage{tikz} \usepackage{amsfonts} \begin{document} \begin{tikzpicture}[node distance=2cm] \node (Q) {$\mathbb{Q}$}; \node (Q6) [above of=Q] {$\mathbb{Q}(\sqrt{6})$}; \node (Q2) [right of=Q6] {$\mathbb{Q}(\sqrt{2})$}; \node (Q3) [left of=Q6] {$\mathbb{Q}(\sqrt{3})$}; \node (Q23) [above of=Q6] {$\mathbb{Q}(\sqrt{2}, \sqrt{3})$}; \node (1f) [right of=Q2] {$\{1, f\}$}; \node (1fg) [right of=1f] {$\{1, fg\}$}; \node (1g) [right of=1fg] {$\{1, g\}$}; \node (G) [below of=1fg] {$\{1
244.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
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class="nb">, f, g, fg\}$}; \node (1) [above of=1fg] {$\{1\}$}; \draw (Q) -- (Q2); \draw (Q) -- (Q3); \draw (Q) -- (Q6); \draw (Q2) -- (Q23); \draw (Q3) -- (Q23); \draw (Q6) -- (Q23); \draw (G) -- (1f); \draw (G) -- (1fg); \draw (G) -- (1g); \draw (1f) -- (1); \draw (1fg) -(1); \draw (1g) -- (1); \end{tikzpicture} \end{document} Original artist: Self • File:Lattice_torsion_points.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d5/Lattice_torsion_points.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Sam Derbyshire • File:Merge-arrows.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/52/Merge-arrows.svg License: Public domain Contributors: ? Original artist: ? • File:MinDiscriminantsCoclass2Type33Sporadic.tiff Source: https://upload.wikimedia.org/wikipedia/en/b/be/MinDiscriminantsCoclass2Type33Sporadic. tiff License: CC0 Contributors: ? Original artist: ? • File:MinDiscriminantsCoclass2Type55Sporadic.png Source: https://upload.wikimedia.org/wikipedia/en/3/32/MinDiscriminantsCoclass2Type55Sporadic. png License: CC0 Contributors: ? Original artist: ? • File:MinDiscriminantsCoclass2Type77Sporadic.tiff Source: https://upload.wikimedia.org/wikipedia/en/a/a0/MinDiscriminantsCoclass2Type77Sporadic. tiff License: CC0 Contributors: ? Original artist: ? • File:MinDiscriminantsTreeQ.png Source: https://upload.wikimedia.org/wikipedia/en/4/4b/MinDiscriminantsTreeQ.png License: CC0 Contributors: ? Original artist: ? • File:MinDiscriminantsTreeU.png Source: https://upload.wikimedia.org/wikipedia/en/1/1d/MinDiscriminantsTreeU.png License: CC0 Contributors: ? Original artist: ? • File:Multiplication_intercept_theorem.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/95/Multiplication_intercept_ theorem.svg License: Public domain Contributors: Own work Original artist: ZooFari • File:Newton-polygon.gif Source: https://upload.wikimedia.org/wikipedia/commons/7/7b/Newton-polygon.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Jathd • File:Number-line.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/93/Number-line.svg License: CC0 Contributors: Own work Original artist: Hakunamenta • File:Number_theory_symbol.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/01/Number_theory_symbol.svg License: CC BY-SA 2.5 Contributors: SVG conversion of Nts.png Original artist: , previous versions by and • File:Nuvola_apps_edu_mathematics_blue-p.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_ mathematics_blue-p.svg License: GPL Contributors: Derivative work from Image:Nuvola apps edu mathematics.png and Image:Nuvola apps edu mathematics-p.svg Original artist: David Vignoni (original icon); Flamurai (SVG convertion); bayo (color) • File:OEISicon_light.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d8/OEISicon_light.svg License: Public domain Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) • File:One5Root.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/40/One5Root.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Loadmaster (David R. Tribble)
• File:PlotDiscriminantsOfComplexCubicFields.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e3/PlotDiscriminantsOfComplexCubicFiel svg License: Public domain Contributors: Own work Original artist: RobHar • File:PlotDiscriminantsOfRealCubicFields.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/32/PlotDiscriminantsOfRealCubicFields. svg License: Public domain Contributors: Own work Original artist: RobHar • File:Punktraster.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ea/Punktraster.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Exxu • File:Quartic_Discriminant.png Source: https://upload.wikimedia.org/wikipedia/commons/2/2b/Quartic_Discriminant.png License: CC BY-SA 4.0 Contributors: Own work Original artist: Salix alba • File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0 Contributors: Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist: Tkgd2007 • File:Rational_Representation.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9c/Rational_Representation.svg License: Public domain Contributors: Own work Original artist: TomT0m • File:Relatives_Numbers_Representation.png Source: https://upload.wikimedia.org/wikipedia/commons/2/27/Relatives_Numbers_ Representation.png License: Public domain Contributors: Own work Original artist: Thomas Douillard, thomas.douillard gmail.com, with “asymptote” • File:Rubik’{}s_cube_v3.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b6/Rubik%27s_cube_v3.svg License: CCBY-SA-3.0 Contributors: Image:Rubik’{}s cube v2.svg Original artist: User:Booyabazooka, User:Meph666 modified by User:Niabot • File:Schematic_depiction_of_ramification.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/44/Schematic_depiction_ of_ramification.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Jakob.scholbach • File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_ with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svg from the Tango project. Original artist: Benjamin D. Esham (bdesham)
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• File:TotallyReal.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/06/TotallyReal.svg License: Public domain Contributors: Own work Original artist: Chenxlee • File:TreeCoclass2RootQ.png Source: https://upload.wikimedia.org/wikipedia/en/1/18/TreeCoclass2RootQ.png License: CC0 Contributors: ? Original artist: ? • File:U+211A.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/07/U%2B211A.svg License: Public domain Contributors: Transferred from en.wikipedia; transferred to Commons by User:Common Good using CommonsHelper. Original artist: Original uploader was Joey-das-WBF at en.wikipedia • File:Wiki_letter_w.svg Source: https://upload.wikimedia.org/wikipedia/en/6/6c/Wiki_letter_w.svg License: Cc-by-sa-3.0 Contributors: ? Original artist: ? • File:Wiki_letter_w_cropped.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Wiki_letter_w_cropped.svg License: CC-BY-SA-3.0 Contributors: • Wiki_letter_w.svg Original artist: Wiki_letter_w.svg: Jarkko Piiroinen • File:Wikibooks-logo-en-noslogan.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan. svg License: CC BY-SA 3.0 Contributors: Own work Original artist: User:Bastique, User:Ramac et al. • File:Wikisource-logo.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg License: CC BY-SA 3.0 Contributors: Rei-artur Original artist: Nicholas Moreau • File:Wiktionary-logo-en.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Wiktionary-logo-en.svg License: Public domain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk · contribs), based on original logo tossed together by Brion Vibber
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