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© 1990

Cyclotomic Fields I and II

Textbook

Part of the Graduate Texts in Mathematics book series (GTM, volume 121)

Table of contents

  1. Front Matter
    Pages i-xvii
  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. Serge Lang
    Pages 244-268
  9. Serge Lang
    Pages 269-279
  10. Serge Lang
    Pages 280-294
  11. Serge Lang
    Pages 295-313
  12. Serge Lang
    Pages 314-328
  13. Serge Lang
    Pages 329-359
  14. Serge Lang
    Pages 360-380
  15. Serge Lang
    Pages 381-396
  16. Back Matter
    Pages 397-436

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 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.

Keywords

Cohomology Prime algebra finite field homomorphism number theory

Authors and affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Bibliographic information

  • Book Title Cyclotomic Fields I and II
  • Authors Serge Lang
  • Series Title Graduate Texts in Mathematics
  • DOI https://doi.org/10.1007/978-1-4612-0987-4
  • Copyright Information Springer-Verlag New York, Inc. 1990
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-387-96671-7
  • Softcover ISBN 978-1-4612-6972-4
  • eBook ISBN 978-1-4612-0987-4
  • Series ISSN 0072-5285
  • Edition Number 2
  • Number of Pages XVII, 436
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published as volumes 59 and 69 in the same series
  • Topics Number Theory
  • Buy this book on publisher's site
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