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Cyclotomic fields I and II / Serge Lang; with an appendix by Karl Rubin

Main Author Lang, Serge, 1927-2005 Secondary Author Rubin, Karl Country Estados Unidos. Edition Combined 2nd ed Publication New York : Springer-Verlag, cop. 1990 Description XVII, 433 p. ; 25 cm Series Graduate texts in mathematics , 121) ISBN 0-387-96671-4 CDU 511.236
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Item type Current location Call number Status Date due Barcode Item holds
Monografia Biblioteca Geral da Universidade do Minho
BGUMD 107576 Available 208594
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Enhanced descriptions from Syndetics:

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.

Table of contents provided by Syndetics

  • Character Sums
  • Stickelberger Ideals and Bernoulli Distributions
  • Complex Analytic Class Number Formulas
  • The p-adic L-function
  • Iwasawa Theory and Ideal Class Groups
  • Kummer Theory over Cyclotomic Zp-extensions
  • Iwasawa Theory of Local Units
  • Lubin-Tate Theory
  • Explicit Reciprocity Laws
  • Measures and Iwasawa Power Series
  • The Ferrero-Washington Theorems
  • Measures in the Composite Case
  • Divisibility of Ideal Class Numbers
  • p-adic Preliminaries
  • The Gamma Function and Gauss Sums
  • Gauss Sums and the Artin-Schreier Curve
  • Gauss Sums as Distributions
  • Appendix: The Main Conjecture
  • Bibliography
  • Index

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