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
