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Graduate Texts in Mathematics

Cyclotomic Fields

Authors: Lang, S.

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About this Textbook

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.

Table of contents (9 chapters)

Table of contents (9 chapters)

Buy this book

eBook $64.99
price for USA in USD (gross)
  • ISBN 978-1-4612-9945-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $84.99
price for USA in USD
  • ISBN 978-1-4612-9947-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Cyclotomic Fields
Authors
Series Title
Graduate Texts in Mathematics
Series Volume
59
Copyright
1978
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag, New York Inc.
eBook ISBN
978-1-4612-9945-5
DOI
10.1007/978-1-4612-9945-5
Softcover ISBN
978-1-4612-9947-9
Series ISSN
0072-5285
Edition Number
1
Number of Pages
253
Topics