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The Theory of Classical Valuations

  • Book
  • © 1999

Overview

Authors:
  1. Paulo Ribenboim

    1. Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

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  • Area of great interest where there is no book to serve as a first introduction to the topic * Fully up-to-date as the author draws upon his long-standing experience * Ribenboim has written other successful Springer titles * Fills a niche in the literature and will serve as the ultimate resource on the subject

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

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    • Paulo Ribenboim
    Pages 1-1

  3. Absolute Values of Fields

    • Paulo Ribenboim
    Pages 3-54

  4. Valuations of a Field

    • Paulo Ribenboim
    Pages 55-78

  5. Polynomials and Henselian Valued Fields

    • Paulo Ribenboim
    Pages 79-105

  6. Extensions of Valuations

    • Paulo Ribenboim
    Pages 107-125

  7. Uniqueness of Extensions of Valuations and Poly-Complete Fields

    • Paulo Ribenboim
    Pages 127-150

  8. Extensions of Valuations: Numerical Relations

    • Paulo Ribenboim
    Pages 151-184

  9. Power Series and the Structure of Complete Valued Fields

    • Paulo Ribenboim
    Pages 185-212

  10. Decomposition and Inertia Theory

    • Paulo Ribenboim
    Pages 213-226

  11. Ramification Theory

    • Paulo Ribenboim
    Pages 227-261

  12. Valuation Characterizations of Dedekind Domains

    • Paulo Ribenboim
    Pages 263-306

  13. Galois Groups of Algebraic Extensions of Infinite Degree

    • Paulo Ribenboim
    Pages 307-331

  14. Ideals, Valuations, and Divisors in Algebraic Extensions of Infinite Degree over ℚ

    • Paulo Ribenboim
    Pages 333-364

  15. A Glimpse of Krull Valuations

    • Paulo Ribenboim
    Pages 365-372

  16. Back Matter

    Pages 373-403
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Keywords

About this book

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.

Reviews

"It is well written, encyclopedic, and authoritative and probably belongs on the shelf of any commutative algebraist or algebraic number theorist."--MATHEMATICAL REVIEWS

Authors and Affiliations

  • Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

    Paulo Ribenboim

Bibliographic Information

  • Book Title: The Theory of Classical Valuations

  • Authors: Paulo Ribenboim

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-0551-7

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1999

  • Hardcover ISBN: 978-0-387-98525-1Published: 21 May 1999

  • Softcover ISBN: 978-1-4612-6814-7Published: 04 October 2012

  • eBook ISBN: 978-1-4612-0551-7Published: 06 December 2012

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XI, 403

  • Topics: K-Theory

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