Introduction to Cyclotomic Fields

Lawrence C. Washington (auth.)9780387906225, 9783540906223, 0387906223, 3540906223

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.

---

Table of contents :
Front Matter….Pages i-xiv
Fermat’s Last Theorem….Pages 1-8
Basic Results….Pages 9-19
Dirichlet Characters….Pages 20-29
Dirichlet L -series and Class Number Formulas….Pages 30-46
p -adic L -functions and Bernoulli Numbers….Pages 47-86
Stickelberger’s Theorem….Pages 87-112
Iwasawa’s Construction of p -adic L -functions….Pages 113-142
Cyclotomic Units….Pages 143-166
The Second Case of Fermat’s Last Theorem….Pages 167-184
Galois Groups Acting on Ideal Class Groups….Pages 185-204
Cyclotomic Fields of Class Number One….Pages 205-231
Measures and Distributions….Pages 232-263
Iwasawa’s Theory of ℤ-extensions….Pages 264-320
The Kronecker—Weber Theorem….Pages 321-331
The Main Conjecture and Annihilation of Class Groups….Pages 332-372
Miscellany….Pages 373-390
Back Matter….Pages 391-490