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.
Author(s): Lawrence C. Washington (auth.)
Series: Graduate Texts in Mathematics 83
Edition: 2
Publisher: Springer-Verlag New York
Year: 1997
Language: English
Pages: 490
City: New York
Tags: Number Theory
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
