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Cyclotomic Fields

  • Serge Lang

Part of the Graduate Texts in Mathematics book series (GTM, volume 59)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Serge Lang
    Pages 1-25
  3. Serge Lang
    Pages 94-122
  4. Serge Lang
    Pages 123-147
  5. Serge Lang
    Pages 166-189
  6. Serge Lang
    Pages 190-219
  7. Serge Lang
    Pages 220-243
  8. Back Matter
    Pages 244-256

About this book

Introduction

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.

Keywords

Fields Kreiskörper Prime algebra finite field homomorphism number theory

Authors and affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-9945-5
  • Copyright Information Springer-Verlag New York 1978
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-9947-9
  • Online ISBN 978-1-4612-9945-5
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site