Cyclotomic Fields I and II

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Springer Science & Business Media, 06.12.2012 - 436 Seiten
Kummer'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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Inhalt

CHAPTER
1
Stickelbergers Theorem
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
Bibliography
421
217
422
69
423
242
425
81
426
84
428
89
431
186
432
291
433
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