## 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 | 3 |

V | 11 |

VII | 22 |

IX | 32 |

XI | 49 |

XIII | 57 |

XIV | 61 |

XV | 65 |

L | 234 |

LII | 239 |

LIII | 254 |

LIV | 266 |

LVI | 267 |

LVII | 271 |

LVIII | 279 |

LIX | 287 |

XVI | 72 |

XVII | 79 |

XVIII | 89 |

XX | 95 |

XXI | 102 |

XXII | 104 |

XXIII | 109 |

XXIV | 115 |

XXVII | 119 |

XXVIII | 127 |

XXIX | 130 |

XXX | 132 |

XXXI | 145 |

XXXIII | 153 |

XXXIV | 155 |

XXXV | 161 |

XXXVI | 169 |

XXXIX | 175 |

XL | 187 |

XLII | 190 |

XLIII | 198 |

XLIV | 207 |

XLV | 208 |

XLVI | 213 |

XLVII | 219 |

XLVIII | 223 |

XLIX | 230 |

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### Common terms and phrases

A-module abelian extension abelian ﬁelds abelian group abelian number ﬁelds algebraic number assume Bernoulli numbers Chapter class ﬁeld theory class number formula coefﬁcients completes the proof congruence conjugation Corollary corresponding cyclic cyclotomic ﬁelds cyclotomic units deﬁned deﬁnition degree denote Dirichlet character distribution divides element exact sequence exists factors ﬁnd ﬁnite ﬁnite abelian ﬁnite index ﬁrst ﬁxed ﬁeld follows easily function ﬁelds Galois group Gauss sums group ring hence ideal class group implies inﬁnite irregular primes isomorphism Kummer Last Theorem Let H lwasawa Main Conjecture Math maximal modp modules nontrivial norm Note Number Theory obtain p-adic L-functions p-adiques polynomial power series prime ideal proof of Lemma proof of Theorem prove pth power quadratic ﬁelds reine angew relations relatively prime result roots of unity satisﬁes Show Stickelberger ideal subgroup Suppose surjective totally ramiﬁed totally real trivial Univ unramiﬁed yields zeta function ZP-extension

### Popular passages

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

Page 459 - On the growth of the first factor of the class number of the prime cyclotomic field...

Page 424 - On a cyclic determinant and the first factor of the class number of the cyclotomic field, ks.

Page 481 - The determination of the imaginary abelian number fields with class number one.

Page 481 - On the rank of the p-divisor class group of Galois extensions of algebraic number fields. Kumamoto J.