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Introduction to Cyclotomic Fields
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This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat's Last Theorem, and Iwasawa's theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott's proof of the vanishing of Iwasawa's f-invariant.
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Table of Contents

1 Fermat's Last Theorem.- 2 Basic Results.- 3 Dirichlet Characters.- 4 Dirichlet L-series and Class Number Formulas.- 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.- 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.- 7 Iwasawa'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. = 0.- 8 Cyclotomic Units.- 8.1. Cyclotomic units.- 8.2. Proof of the p-adic class number formula.- 8.3. Units of $$ mathbb{Q}left( {{zeta _p}} right)$$ and Vandiver's conjecture.- 8.4. p-adic expansions.- 9 The Second Case of Fermat's Last Theorem.- 9.1. The basic argument.- 9.2. The theorems.- 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.- 11 Cyclotomic Fields of Class Number One.- 11.1. The estimate for even characters.- 11.2. The estimate for all characters.- 11.3. The estimate for hm-.- 11.4. Odlyzko's bounds on discriminants.- 11.5. Calculation of hm+.- 12 Measures and Distributions.- 12.1. Distributions.- 12.2. Measures.- 12.3. Universal distributions.- 13 Iwasawa's Theory of $$ {mathbb{Z}_p} -$$ 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 unramified outside p.- 13.6. The main conjecture.- 13.7. Logarithmic derivatives.- 13.8. Local units modulo cyclotomic units.- 14 The Kronecker-Weber Theorem.- 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.- 16 Miscellany.- 16.1. Primality testing using Jacobi sums.- 16.2. Sinnott's proof that = 0.- 16.3. The non-p-part of the class number in a $$ {mathbb{Z}_p} -$$ extension.- 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.- List of Symbols.

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