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The Theory of Classical Valuations

  • Paulo Ribenboim

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Paulo Ribenboim
    Pages 1-1
  3. Paulo Ribenboim
    Pages 3-54
  4. Paulo Ribenboim
    Pages 55-78
  5. Paulo Ribenboim
    Pages 79-105
  6. Paulo Ribenboim
    Pages 107-125
  7. Paulo Ribenboim
    Pages 151-184
  8. Paulo Ribenboim
    Pages 213-226
  9. Paulo Ribenboim
    Pages 227-261
  10. Paulo Ribenboim
    Pages 263-306
  11. Paulo Ribenboim
    Pages 365-372
  12. Back Matter
    Pages 373-403

About this book

Introduction

In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive p-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of p-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the p-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book.

Keywords

Algebra Division Finite Topology commutative property function fundamental theorem theorem

Authors and affiliations

  • Paulo Ribenboim
    • 1
  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-0551-7
  • Copyright Information Springer Science+Business Media New York 1999
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-6814-7
  • Online ISBN 978-1-4612-0551-7
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site