## 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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### Contents

III | 1 |

IV | 9 |

V | 20 |

VI | 30 |

VII | 47 |

VIII | 55 |

IX | 59 |

X | 63 |

XXXVIII | 232 |

XXXIX | 237 |

XL | 252 |

XLI | 264 |

XLII | 265 |

XLIII | 269 |

XLIV | 277 |

XLV | 285 |

XI | 70 |

XII | 77 |

XIII | 87 |

XIV | 93 |

XV | 100 |

XVI | 102 |

XVII | 107 |

XVIII | 113 |

XIX | 117 |

XX | 125 |

XXI | 128 |

XXII | 130 |

XXIII | 143 |

XXIV | 151 |

XXV | 153 |

XXVI | 159 |

XXVII | 167 |

XXVIII | 173 |

XXIX | 185 |

XXX | 188 |

XXXI | 196 |

XXXII | 205 |

XXXIII | 206 |

XXXIV | 211 |

XXXV | 217 |

XXXVI | 221 |

XXXVII | 228 |

### Common terms and phrases

A-module abelian extension abelian group annihilates assume Bernoulli numbers Chapter character of conductor class field theory class number formula CM-field coefficients completes the proof congruence conjugation Corollary corresponding cyclic cyclotomic fields cyclotomic units cyclotomic Zp-extension defined degree denote Dirichlet characters distribution divides elements exact sequence exists factors Fermat's Last Theorem finite abelian finite index follows easily Galois group hence ideal class group implies isomorphism Iwasawa Kummer Let F logp Main Conjecture Math maximal modp modulo multiplicative nontrivial norm Note Number Theory obtain odd prime p-adic L-functions polynomial power series prime ideal prime power proof of Lemma proof of Proposition proof of Theorem prove pth power Q(Cm Q(CP rational integer relation relatively prime result roots of unity satisfies Show Stickelberger subfield subgroup Suppose surjective totally ramified totally real trivial unique unramified Vandiver's conjecture yields zeta function

### Popular passages

Page iii - PREFACE TO THE SECOND EDITION Since the publication of the first edition of this book, "Definitions of Electrical Terms...

Page 453 - On the growth of the first factor of the class number of the prime cyclotomic field...

Page 475 - The determination of the imaginary abelian number fields with class number one.

Page 475 - On the rank of the p-divisor class group of Galois extensions of algebraic number fields. Kumamoto J.

Page 434 - On the trace of the ring of integers of an abelian number field, Acta Arith., 62, 1992, 383-389.