Introduction to Cyclotomic Fields

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 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. µ = 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 cyclotomicunits. - 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 µ = 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.