## Cyclotomic Fields I and IIKummer'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 11] 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 |

Changing the Prime | 200 |

The Reciprocity | 203 |

The Kummer Pairing | 204 |

The Logarithm | 211 |

Application of the Logarithm to the Local Symbol | 217 |

CHAPTER 9 | 222 |

Statement of the Reciprocity Laws | 223 |

The Logarithmic Derivative | 224 |

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 |

Universal Distributions | 57 |

The DavenportHasse Distribution | 61 |

Appendix Distributions | 65 |

Complex Analytic Class Number Formulas 1 Gauss Sums on Zm2 | 69 |

Primitive Lseries | 72 |

Decomposition of Lseries | 75 |

The +1eigenspaces | 81 |

Cyclotomic Units | 84 |

The Dedekind Determinant | 89 |

Bounds for Class Numbers CHAPTER 4 | 91 |

Measures and Power Series | 95 |

Operations on Measures and Power Series | 101 |

The Mellin Transform and padic Lfunction | 105 |

Appendix The padic Logarithm | 111 |

The padic Regulator | 112 |

The Formal Leopoldt Transform | 115 |

The padic Leopoldt Transform | 117 |

CHAPTER 5 | 126 |

Weierstrass Preparation Theorem | 129 |

Modules over ZpX | 131 |

Zpextensions and Ideal Class Groups | 137 |

The Maximal pabelian pramified Extension | 146 |

The Galois Group as Module over the Iwasawa Algebra | 147 |

CHAPTER 6 | 154 |

Kummer Theory over Cyclotomic Zoextensions 1 The Cyclotomic Zpextension 2 The Maximal pabelian pramified Extension of the Cyclotomic Zpe... | 155 |

Cyclotomic Units as a Universal Distribution | 157 |

CHAPTER 8 | 159 |

The IwasawaLeopoldt Theorem and the KummerVandiver Conjecture CHAPTER 7 | 160 |

Iwasawa Theory of Local Units | 166 |

Projective Limit of the Unit Groups | 175 |

A Basis for UX over | 179 |

The CoatesWiles Homomorphism | 182 |

The Closure of the Cyclotomic Units | 186 |

LubinTate Theory 1 LubinTate Groups | 190 |

Formal padic Multiplication | 196 |

A Local Pairing with the Logarithmic Derivative | 229 |

The Main Lemma for Highly Divisible x and 0 | 233 |

The Main Theorem for the Symbol x xnn | 237 |

The Main Theorem for Divisible x and 0 unit | 239 |

End of the Proof of the Main Theorems | 242 |

CHAPTER 10 | 246 |

Iwasawa Invariants for Measures | 247 |

Application to the Bernoulli Distributions | 251 |

Class Numbers as Products of Bernoulli Numbers | 258 |

Probabilities | 261 |

Washingtons Theorem | 265 |

CHAPTER 11 | 270 |

Basic Lemma and Applications | 271 |

Equidistribution and Normal Families | 272 |

An Approximation Lemma | 276 |

Proof of the Basic Lemma | 277 |

CHAPTER 12 | 282 |

Measures and Power Series in the Composite Case | 283 |

The Associated Analytic Function on the Formal Multiplicative Group | 286 |

Computation of Lp1 y in the Composite Case Contents | 291 |

CHAPTER 13 | 296 |

CHAPTER 14 | 314 |

Analytic Representation of Roots of Unity | 323 |

CHAPTER 15 | 330 |

The Frobenius Endomorphism | 338 |

padic Banach Spaces | 348 |

CHAPTER 16 | 362 |

CHAPTER 17 | 382 |

200 | 393 |

APPENDIX BY KARL RUBIN | 397 |

The Ideal Class Group of Qup | 403 |

Proof of Theorem 5 1 | 411 |

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### Common terms and phrases

A-module A(pm assume automorphism Banach basis Banach space Bernoulli numbers Bernoulli polynomials Chapter class field theory class number CM field coefficients commutative concludes the proof conductor congruence Corollary cyclic cyclotomic fields cyclotomic units define denote det(I Dirichlet character distribution relation divisible Dwork eigenspace eigenvalue elements endomorphism extension factor follows formal group formula Frobenius Frobenius endomorphism Galois group Gauss sums gives group ring Hence homomorphism ideal class group isomorphism kernel KUBERT Kummer Leopoldt Let F linear mod 7t module multiplicative group norm notation number field odd characters p-unit polynomial positive integer power series associated prime number primitive projective limit Proposition proves the lemma proves the theorem Q(up quasi-isomorphism rank right-hand side root of unity satisfies shows subgroup suffices to prove Suppose surjective Theorem 3.1 trivial unique unramified values vector write zeta function Zp-extension