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
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