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Author Washington, Lawrence C. author Introduction to Cyclotomic Fields [electronic resource] / by Lawrence C. Washington New York, NY : Springer New York : Imprint: Springer, 1997 Second Edition http://dx.doi.org/10.1007/978-1-4612-1934-7 XIV, 490 p. online resource

SUMMARY

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

CONTENT

1 Fermatโ{128}{153}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 Stickelbergerโ{128}{153}s Theorem -- 6.1. Gauss sums -- 6.2. Stickelbergerโ{128}{153}s theorem -- 6.3. Herbrandโ{128}{153}s theorem -- 6.4. The index of the Stickelberger ideal -- 6.5. Fermatโ{128}{153}s Last Theorem -- 7 Iwasawaโ{128}{153}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. ยต = 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 Vandiverโ{128}{153}s conjecture -- 8.4. p-adic expansions -- 9 The Second Case of Fermatโ{128}{153}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 Vandiverโ{128}{153}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. Odlyzkoโ{128}{153}s bounds on discriminants -- 11.5. Calculation of hm+ -- 12 Measures and Distributions -- 12.1. Distributions -- 12.2. Measures -- 12.3. Universal distributions -- 13 Iwasawaโ{128}{153}s Theory of $${\mathbb{Z}_p} -$$ extensions -- 13.1. Basic facts -- 13.2. The structure of A-modules -- 13.3. Iwasawaโ{128}{153}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 Kroneckerโ{128}{148}Weber Theorem -- 15 The Main Conjecture and Annihilation of Class Groups -- 15.1. Stickelbergerโ{128}{153}s theorem -- 15.2. Thaineโ{128}{153}s theorem -- 15.3. The converse of Herbrandโ{128}{153}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. Sinnottโ{128}{153}s proof that ยต = 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

Mathematics Number theory Mathematics Number Theory

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