## Introduction to cyclotomic fieldsIntroduction 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. |

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A-module abelian extension abelian group assume Bernoulli numbers Chapter class field theory class number formula CM-field coefficients completes the proof congruence Corollary corresponding cyclic cyclotomic fields cyclotomic units cyclotomic Zp-extension defined degree denote Dirichlet character distribution divides elements example Exercise fact factors Fermat's Last Theorem finite index fixed field follows easily Galois group hence ideal class group idempotents inertia group isomorphism Iwasawa Kummer Let H logp Lp(l maximal mod p2 mod(l mod(p modulo nontrivial Note number field obtain p-adic L-functions p-rank p-Sylow subgroup polynomial power series prime ideal prime power proof of Lemma proof of Theorem prove pth power mod Q(Cm Q(CP quadratic rational integer mod relatively prime result root of unity Show Stickelberger Suppose surjective totally ramified totally real trivial unique unramified Vandiver's conjecture Z/pZ zeta function