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" Introduction to Cyclotomic Fields "
by Lawrence C. Washington.
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BL
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| Record Number
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573616
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b402835
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| Main Entry
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Washington, Lawrence C.
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| Title & Author
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Introduction to Cyclotomic Fields\ by Lawrence C. Washington.
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| Edition Statement
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Second Edition.
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| Publication Statement
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New York, NY :: Springer New York :: Imprint: Springer,, 1997.
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| Series Statement
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Graduate Texts in Mathematics,; 83
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| ISBN
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9781461219347
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: 9781461273462
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| Contents
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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. [micro] = 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 [micro] = 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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| Abstract
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Introduction to Cyclotomic Fields is a carefully written exposition of a central area of number theory that can be used as a second course in algebraic number theory. Starting at an elementary level, the volume covers p-adic L-functions, class numbers, cyclotomic units, Fermat's Last Theorem, and Iwasawa's theory of Z_p-extensions, leading the reader to an understanding of modern research literature. Many exercises are included. The second edition includes a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture. There is also a chapter giving other recent developments, including primality testing via Jacobi sums and Sinnott's proof of the vanishing of Iwasawa's f-invariant.
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Mathematics.
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Number theory.
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SpringerLink (Online service)
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