《割圓域導論（第2版）》是2014年世界圖書出版公司出版的著作，作者是[美] La（美）wrence C. Washington 。
基本介紹
- 中文名：《割圓域導論（第2版）》
- 作者：[美]La（美）wrence C. Washington
- 出版社：世界圖書出版公司
- 出版時間：2014年07月01日
- ISBN：9787510077852
內容簡介,目錄,
內容簡介
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.
目錄
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 7
lwasawa's Construction of p-adic L-functions
7.1. Group rings and power series
7.2. p-adic L-functions
7.3. Applications
7.4. Function fields
7.5. μ=O
CHAPTER 8
Cyclotomic Units
8.1. Cyclotomic units
8.2. Proof of the p-adic class number formula
8.3. Units of O(Cp) 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 1
Cyclotomic Fields of Class Number One
11.1. The estimate for even characters
l1.2. The estimate for all characters
11.3. The estimate for hm,
11.4. Odlyzko's bounds on discriminants
11.5. Calculation of hm+
CHAPTER 12
Measures and Distributions
12.1. Distributions
12.2. Measures
12.3. Universal distributions
CHAPTER 13
Iwasawa's Theory of Zp-extensions
13.1. Basic facts
13.2. The structure of A-modules
13.3. Iwasawa's theorem
13.4. Consequences
13.5. The maximal abelian p-extension unramifiexl outside p
13.6. The main conjecture
13.7. Logarithmic derivatives
13.8. Local units modulo cyclotomi~ units
CHAPTER 14
The Kronecker-Weber Theorem
CHAPTER 15
The Main Conjecture and Annihilation of Class Groups
15.1. Stickelberger's theorem
15.2. Thaine's theorem
15.3. The converse of Herbrand's theorem
15.4. The Main Conjecture
15.5. Adjoints
15.6. Technical results from Iwasawa theory
15.7. Proof of the Main Conjecture
CHAPTER 16
Misccllany
16.1. Primality testing using Jacobi sums
16.2. Sinnott's proof thatμ= 0
16.3. The non-p-part of the class number in a Zp-extension
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
