割圆域导论（第二版 英文版） 出版时间：2014年版 内容简介 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. 目录 Preface to the Second Edition Preface to the First Edition CHAPTER Ⅰ 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 6.1. Gauss sums 6.2. Stickelberger's theorem 6.3. Herbrand's theorem 6.4. The index of the Stickelberger ideal 6.5. Fermat's Last Theorem CHAPTER ? lwasawa's Construction of p-adic L-functions 7.1. Group tings and power series 7.2. p-adic L-functions 7.3. Appfications 7.4. Function fields 7.5. CHAPTER 8 Cyclotomic Units 8.1. Cyclotomic units 8.2. Proof of the p-adic class number formula 8.3. Units of O(~,) and Vandiver's conjecture 8.4. p-adic expansions CHAPTER 9 The Second Case of Fermat's Last Theorem 9.1. The basic argument 9.2. The theorems CHAPTER 10 Galois Groups Acting on Ideal Class Groups 10.1. Some theorems on class groups 10.2. Reflection theorems 10.3. Consequences of Vandiver's conjecture CHAPTER I ! Cyclotomic Fields of Class Number One 11.1. The estimate for even characters 11.2. The estimate for all characters …… CHAPTER 14 CHAPTER 15 The Main Conjecture and Annihilation of Class Groups CHAPTER 16 Miscellany Appendix 1. Inverse limits 2. Infinite Galois theory and ramification theory 3. Class field theory Tables 1. Bernoulli numbers 2. Irregular primes 3. Relative class numbers 4. Real class numbers Bibliography List of Symbols Index
