## Cyclotomic FieldsKummer's work on cyclotomic fields paved the way for the development of algebraic number theory in general by Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. However, the success of this general theory has tended to obscure special facts proved by Kummer about cyclotomic fields which lie deeper than the general theory. For a long period in the 20th century this aspect of Kummer's work seems to have been largely forgotten, except for a few papers, among which are those by Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. In the mid 1950's, the theory of cyclotomic fields was taken up again by Iwasawa and Leopoldt. Iwasawa viewed cyclotomic fields as being analogues for number fields of the constant field extensions of algebraic geometry, and wrote a great sequence of papers investigating towers of cyclotomic fields, and more generally, Galois extensions of number fields whose Galois group is isomorphic to the additive group of p-adic integers. Leopoldt concentrated on a fixed cyclotomic field, and established various p-adic analogues of the classical complex analytic class number formulas. In particular, this led him to introduce, with Kubota, p-adic analogues of the complex L-functions attached to cyclotomic extensions of the rationals. Finally, in the late 1960's, Iwasawa [Iw 1 I] . made the fundamental discovery that there was a close connection between his work on towers of cyclotomic fields and these p-adic L-functions of Leopoldt-Kubota. |

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### Contents

1 | |

6 | |

Relations in the Ideal Classes | 14 |

Jacobi Sums as Hecke Characters | 16 |

Gauss Sums Over Extension Fields | 20 |

Application to the Fermat Curve | 22 |

CHAPTER 2 | 26 |

The Index of the First Stickelberger Ideal | 27 |

Universal Distributions | 57 |

The DavenportHasse Distribution | 61 |

Complex Analytic Class Number Formulas | 69 |

Primitive Lseries | 72 |

i | 92 |

CHAPTER 5 | 126 |

Modules over ZpX | 133 |

The Maximal pabelian pramified Extension | 146 |

Bernoulli Numbers | 32 |

Integral Stickelberger Ideals | 43 |

General Comments on Indices | 48 |

The Index for k Even | 49 |

The Index for k Odd | 50 |

Twistings and Stickelberger Ideals | 51 |

Stickelberger Elements as Distributions | 53 |

CHAPTER 6 | 154 |

Iwasawa Theory of Local Units | 166 |

CHAPTER 8 | 220 |

CHAPTER 9 | 222 |

244 | |

252 | |

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abelian extension algebraic number assume automorphism Bernoulli numbers Bernoulli polynomials Chapter class field theory class number coefficients computation concludes the proof congruence cyclotomic fields cyclotomic units defined mod denote distribution relation elements factor group follows formal group formal multiplicative group formula Frobenius power series Galois group Gauss sum group ring H. W. LEOPOLDT Hence homomorphism ideal class group isomorphism Iwasawa algebra Kummer Let x e Lubin-Tate group Math maximal ideal mod deg mod pn module non-trivial norm notation number field obvious p-adic L-function p-integral p-primary positive integer power series power series associated prime power primitive projective limit proves the theorem Q(um Q(un residue class root of unity satisfies shows Stickelberger ideal suffices to prove Suppose surjective symbol Theorem 2.2 totally ramified trivial unramified values Vandiver conjecture vector write x e A(pm zeta function Zp-extension