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Classical Theory of Algebraic Numbers

  • Paulo Ribenboim

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Introduction

    1. Front Matter
      Pages 1-3
    2. Paulo Ribenboim
      Pages 13-31
  3. Part One

    1. Front Matter
      Pages 33-35
    2. Paulo Ribenboim
      Pages 37-59
    3. Paulo Ribenboim
      Pages 61-81
  4. Part Two

    1. Front Matter
      Pages 83-83
    2. Paulo Ribenboim
      Pages 85-105
    3. Paulo Ribenboim
      Pages 107-121
    4. Paulo Ribenboim
      Pages 123-139
    5. Paulo Ribenboim
      Pages 141-151
    6. Paulo Ribenboim
      Pages 153-166
    7. Paulo Ribenboim
      Pages 167-187
    8. Paulo Ribenboim
      Pages 189-205
    9. Paulo Ribenboim
      Pages 207-232
    10. Paulo Ribenboim
      Pages 273-289
    11. Paulo Ribenboim
      Pages 291-336
  5. Part Three

    1. Front Matter
      Pages 337-337
    2. Paulo Ribenboim
      Pages 339-366
    3. Paulo Ribenboim
      Pages 367-397
    4. Paulo Ribenboim
      Pages 429-460
  6. Part Four

    1. Front Matter
      Pages 461-461
    2. Paulo Ribenboim
      Pages 463-485
    3. Paulo Ribenboim
      Pages 487-503
    4. Paulo Ribenboim
      Pages 505-521
    5. Paulo Ribenboim
      Pages 523-542
    6. Paulo Ribenboim
      Pages 567-593
    7. Paulo Ribenboim
      Pages 595-616
  7. Back Matter
    Pages 665-682

About this book

Introduction

Gauss created the theory of binary quadratic forms in "Disquisitiones Arithmeticae" and Kummer invented ideals and the theory of cyclotomic fields in his attempt to prove Fermat's Last Theorem. These were the starting points for the theory of algebraic numbers, developed in the classical papers of Dedekind, Dirichlet, Eisenstein, Hermite and many others. This theory, enriched with more recent contributions, is of basic importance in the study of diophantine equations and arithmetic algebraic geometry, including methods in cryptography. This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples. The Introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields. Part One is devoted to residue classes and quadratic residues. In Part Two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, inertia and ramification of ideals. Part Three is devoted to Kummer's theory of cyclomatic fields, and includes Bernoulli numbers and the proof of Fermat's Last Theorem for regular prime exponents. Finally, in Part Four, the emphasis is on analytical methods and it includes Dinchlet's Theorem on primes in arithmetic progressions, the theorem of Chebotarev and class number formulas. A careful study of this book will provide a solid background to the learning of more recent topics.

Keywords

algebra algebraic geometry automorphism cryptography diophantine equation field prime number quadratic form

Authors and affiliations

  • Paulo Ribenboim
    • 1
  1. 1.Department of MathematicsQueen’s UniversityKingstonCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-387-21690-4
  • Copyright Information Springer Science+Business Media New York 2001
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-2870-2
  • Online ISBN 978-0-387-21690-4
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site
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