Algebraic Number Theory

"... Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory." Medelingen van het wiskundig genootschap, 1995

Contents

Preface 7

Ch. 1 Basic Number Theory 8

1 Orders in Algebraic Number Fields 9
2 Rings with Divisor Theory 22
3 Dedekind Rings 27
4 Valuations 45
5 Harmonic Analysis on Local and Global Fields 63
6 Hecke L-Series and the Distribution of Prime Ideals 70

Ch. 2 Class Field Theory 90

1 The Main Theorems of Class Field Theory 92
2 Complex Multiplication 107
3 Cohomology of Groups 112
4 Proof of the Main Theorems of Class Field Theory 121
5 Simple Algebras 131
6 Explicit Reciprocity Laws and Symbols 137
7 Further Results of Class Field Theory 145

Ch. 3 Galois Groups 150

1 Cohomology of Profinite Groups 151
2 Galois Cohomology of Local and Global Fields 168
3 Extensions with Given Galois Groups 182

Ch. 4 Abelian Fields 192

1 The Integers of an Abelian Field 193
2 The Arithmetical Class Number Formula 195
3 Iwasawa's Theory of [Gamma]-Extensions 206
4 p-adic L-Functions 211

Ch. 5 Artin L-Functions and Galois Module Structure 219

1 Artin L-Functions 222
2 Galois Module Structure and Artin Root Numbers 234

App. 1 Fields, Domains, and Complexes 237
App. 2 Quadratic Residues 240
App. 3 Locally Compact Groups 241
App. 4 Bernoulli Numbers 243

Tables 245
References 251
Author Index 263
Subject Index 266