Since the publication of the first edition, several remarkable developments have taken place. The work of Thaine, Kolyvagin, and Rubin has produced fairly elementary proofs of Ribet's converse of Herbrand's theorem and of the Main Conjecture. The original proofs of both of these results used delicate techniques from algebraic geometry and were inaccessible to many readers. Also, Sinnott discovered a beautiful proof of the vanishing of Iwasawa's u-invariant that is much simpler than the one given in Chapter 7. Finally, Fermat's Last Theorem was proved by Wiles, using work of Frey, Ribet, Serre, Mazur, Langlands-Tunnell, Taylor-Wiles, and others. Although the proof, which is based on modular forms and elliptic curves, is much different from the cyclotomic approaches described in this book, several of the ingredients were inspired by ideas from cyclotomic fields and Iwasawa theory.

本书为英文版。

Preface to the Second Edition

Preface to the First Edition

CHAPTER 1 Fermat‘s Last Theorem

CHAPTER 2 Basic Results

CHAPTER 3 Dirichlet Characters

CHAPTER 4 Dirichlet L-series and Class Number Formulas

CHAPTER 5 p-adic L-functions and Bernoulli Numbers

5.1. p-adic functions

5.2. p-adic L-functions

5.3. Congruences

5.4. The value at s=1

5.5. The p-adic regulator

5.6. Applications of the class number formula

CHAPTER 6 Stickelberger‘s Theorem

CHAPTER 7 Iwasawa's Construction of p-adic L-function

CHAPTER 8 Cyclotomic Units

CHAPTER 9 The Second Case of Fermat's Last Theorm

CHAPTER 10 Galois Groups Acting on Ideal Class Groups

CHAPTER 11 Cyclotomic Fields of Class Number One

CHAPTER 12 Measures and Distribution

CHAPTER 13 Iwasawa's Theory of Zp-extensions

CHAPTER 14 The Kronecker-Weber Theorem

CHAPTER 15 The Main Conjecture and Annihilation of Class Groups

CHAPTER 16 Misscellany

Appendix

Tables

Bibliography

List of Symbols

Index