Introduction to Cyclotomic Fields [Paperback]

This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermats Last Theorem, and Iwasawas 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 Sinnotts proof of the vanishing of Iwasawas f-invariant.

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.1 Fermats 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 Stickelbergers Theorem.- 6.1. Gauss sums.- 6.2. Stickelbergers theorem.- 6.3. Herbrands theorem.- 6.4. The index of the Stickelberger ideal.- 6.5. Fermats Last Theorem.- 7 Iwasawas 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. Unitsl3&