Cyclotomic Fields

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 253 pages
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 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

Foreword 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
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
Bibliography
244
166
252
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