Class Field Theory

212 Pages·2007·1.123 MB·English

by  Franz Lemmermeyer

Franz Lemmermeyer
Class Field Theory
April 30, 2007
Franz Lemmermeyer
email [email protected]
http://www.rzuser.uni-heidelberg.de/~hb3/
Preface
Class field theory has a reputation of being an extremely beautiful part of
numbertheoryandanextremelydifficultsubjectatthesametime.Forsome-
one with a good background in local fields, Galois cohomology and profinite
groups there exist accounts of class field theory that reach the summit (exis-
tencetheoremsandArtinreciprocity)quitequickly;infactNeukirch’sbooks
show that it is nowadays possible to cover the main theorems of class field
theory in a single semester.
Students who have just finished a standard course on algebraic number
theory,however,rarelyhavethenecessaryfamiliaritywiththemoreadvanced
tools of the trade. They are looking for sources that include motivational
material, routine exercises, problems, and applications.
These notes aim at serving this audience. I have chosen the classical ap-
proach to class field theory for the following reasons:
1. Zeta functions and L-series are an important tool not only in algebraic
number theory, but also in algebraic geometry.
2. The analytic proof of the first inequality is very simple once you know
that the Dedekind zeta function has a pole of order 1 at s=1.
3. The algebraic techniques involved in the classical proof of the second
inequality give us results for free that have to be derived from class field
theory in the idelic approach; among the is the ambiguous class number
formula,Hilbert’sTheorem94,orFurtw¨angler’sprincipalgenustheorem.
4. Many of the central unsolved problems in modern number theory are
directlyconnectedtoanalyticobjects.LetmejustmentiontheRiemann
conjecture for various L-functions, the Stark conjectures, the conjecture
of Birch and Swinnerton-Dyer, and the whole Langlands program.
I also have tried to approach certain central results by first treating special
cases; this is not particularly elegant, but it helps students to see how some
of the more technical proofs evolved from relatively simple considerations.
vi
Table of Contents
Part I. Dirichlet’s Analytic Methods
1. Dirichlet Series for Quadratic Characters ................. 3
1.1 Euler ................................................. 3
1.2 Basic Properties of the Riemann Zeta Function............. 7
1.3 Quadratic Number Fields................................ 12
1.4 Gauss................................................. 13
1.5 Dirichlet’s L-series...................................... 15
2. The Nonvanishing of L(1,χ) for Quadratic Characters .... 21
2.1 Dirichlet’s Proof for Prime Discriminants .................. 21
2.2 Nonvanishing of Dirichlet’s L-functions.................... 27
2.3 Computation of L(1,χ) ................................. 29
3. Primes in Arithmetic Progression......................... 41
3.1 Characters............................................. 42
3.2 Primes in Arithmetic Progression......................... 45
3.3 Cyclotomic Number Fields............................... 47
4. Dirichlet.................................................. 53
4.1 Dirichlet’s L-series for Quadratic Forms ................... 53
4.2 Genus Theory for Quadratic Number Fields................ 55
4.3 Primes with Prescribed Residue Characters ................ 56
4.4 Primes Represented by Binary Quadratic Forms............ 58
5. Algebraic Number Fields ................................. 61
5.1 Archimedean Valuations of a Number Field ................ 61
5.2 Arithmetic of Number Fields............................. 62
5.3 Prime Decomposition in Relative Extensions ............... 65
5.4 Prime Ideals in Galois Extensions ........................ 67
5.5 Minkowski Bounds...................................... 70
6. Dirichlet’s Unit Theorem ................................. 77
6.1 Units in Quadratic Number Fields ........................ 77
6.2 Dirichlet’s Unit Theorem ................................ 79
viii Table of Contents
6.3 The Unit Theorems of Minkowski and Herbrand............ 82
7. Dedekind’s Zeta Function................................. 87
7.1 Distribution of Ideals ................................... 88
7.2 Dirichlet’s Class Number Formula ........................ 95
7.3 Cyclotomic Fields ...................................... 96
8. Density Theorems ........................................ 97
8.1 Kronecker’s Density Theorem ............................ 97
8.2 Frobenius Density Theorem for Abelian Extensions ......... 100
8.3 Kummer Extensions .................................... 102
8.4 Decomposition Laws in Kummer Extensions ............... 105
8.5 Density Theorems of Kummer and Hilbert................. 106
Part II. Hilbert Class Fields
9. The Hilbert Class Field................................... 111
9.1 Weber’s Motivation..................................... 111
√
9.2 The Field Q( −5) ..................................... 114
√
9.3 The Field Q( 3)....................................... 115
9.4 Hilbert Class Field Theory II ............................ 116
10. The First Inequality ...................................... 123
10.1 Weber’s Inequality...................................... 123
10.2 Proof of the First Inequality ............................. 126
10.3 Consequences of the First Inequality ...................... 128
11. The Second Inequality.................................... 133
11.1 Preliminaries........................................... 133
11.2 The Second Inequality for Unramified Extensions........... 136
11.3 The Ambiguous Class Number Formula ................... 138
11.4 The Herbrand Quotient of the Unit Group................. 142
12. Examples of Hilbert Class Fields ......................... 147
13. The Artin Symbol ........................................ 149
13.1 Inertia Groups ......................................... 149
13.2 The Symbols of Frobenius and Artin...................... 150
13.3 The Artin Isomorphism ................................. 153
14. Frobenius Density ........................................ 157
14.1 Frobenius and his Density Theorem....................... 157
14.2 Group Theoretical Preliminaries.......................... 162
14.3 Prime Ideal Decomposition in Nonnormal Extensions ....... 163
14.4 The Proof of Frobenius’ Density Theorem ................. 165
Table of Contents ix
Part III. Takagi’s Class Field Theory
15. Ideal Groups ............................................. 171
15.1 Generalized Class Groups ............................... 171
15.2 Takagi’s Class Field Theory.............................. 175
15.3 The Fundamental Inequalities............................ 180
16. Artin’s Reciprocity Law .................................. 183
16.1 Cyclotomic Fields ...................................... 183
16.2 Base Change........................................... 183
16.3 Proof of Artin’s Reciprocity Law ......................... 183
17. The Existence Theorem .................................. 185
18. Norm Residues and Higher Ramification ................. 187
18.1 Higher Ramification Groups ............................. 187
Part IV. Appendix
A. Gamma, Theta, and Zeta ................................. 193
A.1 Euler’s Gamma Function ................................ 193
A.2 Jacobi’s Theta Functions ................................ 193
A.3 Riemann’s Zeta Function ................................ 193
A.4 Quadratic Gauss Sums .................................. 193
B. A Beginner’s Guide to Galois Cohomology................ 195
B.1 H1(G,A).............................................. 195
B.2 Hb0(G,A).............................................. 195
B.3 Hb−1(G,A)............................................. 195
B.4 Galois Cohomology for Cyclic Groups ..................... 195
B.5 Herbrand’s Lemma ..................................... 195
B.6 Capitulation ........................................... 195
B.7 Ambiguous Ideal Classes ................................ 195
C. Solutions of Selected Problems ........................... 197
Bibliography.................................................. 203
x Table of Contents
Part I
Dirichlet’s Analytic Methods
1