Skip to main content
SpringerLink
Log in

Classical Theory of Algebraic Numbers

  • Textbook
  • © 2001

Overview

Authors:
  1. Paulo Ribenboim

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

    View author publications

    You can also search for this author in PubMed   Google Scholar

Part of the book series: Universitext (UTX)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 69.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (29 chapters)

  1. Front Matter

    Pages i-xxiv
    Download chapter PDF

  2. Introduction

    1. Front Matter

      Pages 1-3
      Download chapter PDF

    2. Unique Factorization Domains, Ideals, and Principal Ideal Domains

      • Paulo Ribenboim
      Pages 5-11

    3. Commutative Fields

      • Paulo Ribenboim
      Pages 13-31

  3. Part One

    1. Front Matter

      Pages 33-35
      Download chapter PDF

    2. Residue Classes

      • Paulo Ribenboim
      Pages 37-59

    3. Quadratic Residues

      • Paulo Ribenboim
      Pages 61-81

  4. Part Two

    1. Front Matter

      Pages 83-83
      Download chapter PDF

    2. Algebraic Integers

      • Paulo Ribenboim
      Pages 85-105

    3. Integral Basis, Discriminant

      • Paulo Ribenboim
      Pages 107-121

    4. The Decomposition of Ideals

      • Paulo Ribenboim
      Pages 123-139

    5. The Norm and Classes of Ideals

      • Paulo Ribenboim
      Pages 141-151

    6. Estimates for the Discriminant

      • Paulo Ribenboim
      Pages 153-166

    7. Units

      • Paulo Ribenboim
      Pages 167-187

    8. Extension of Ideals

      • Paulo Ribenboim
      Pages 189-205

    9. Algebraic Interlude

      • Paulo Ribenboim
      Pages 207-232

    10. The Relative Trace, Norm, Discriminant, and Different

      • Paulo Ribenboim
      Pages 233-257

    11. The Decomposition of Prime Ideals in Galois Extensions

      • Paulo Ribenboim
      Pages 259-272

    12. The Fundamental Theorem of Abelian Extensions

      • Paulo Ribenboim
      Pages 273-289

    13. Complements and Miscellaneous Numerical Examples

      • Paulo Ribenboim
      Pages 291-336
Back to top

Keywords

About this book

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.

Reviews

From the reviews of the second edition:

"This book is a thorough self-contained introduction to algebraic number theory. … The book is aimed at graduate students. The author made a great effort to make the subject easier to understand. The proofs are very detailed, there are plenty of examples and there are exercises at the end of almost all chapters … . The book contains a great amount of material, more than enough for a year-long course." (Gábor Megyesi, Acta Scientiarum Mathematicarum, Vol. 69, 2003)

"There is a wealth of material in this book. The approach is very classical and global. … the author keeps his presentation self-contained. The author has made a real effort to make the book accessible to students. Proofs are given in great detail, and there are many examples and exercises. The book would serve well as a text for a graduate course in classical algebraic number theory." (Lawrence Washington, Mathematical Reviews, Issue 2002 e)

"Ribenboims’s ‘Classical Theory of Algebraic Numbers’ is an introduction to algebraic number theory on an elementary level … . Ribenboim’s book is a well written introduction to classical algebraic number theory … and the perfect textbook for students who need lots of examples." (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1082, 2006)

Authors and Affiliations

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

    Paulo Ribenboim

Bibliographic Information

  • Book Title: Classical Theory of Algebraic Numbers

  • Authors: Paulo Ribenboim

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-0-387-21690-4

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2001

  • Hardcover ISBN: 978-0-387-95070-9Published: 30 March 2001

  • Softcover ISBN: 978-1-4419-2870-2Published: 03 December 2010

  • eBook ISBN: 978-0-387-21690-4Published: 11 November 2013

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 2

  • Number of Pages: XXIV, 682

  • Additional Information: Originally published by John Wiley & Sons, New York, 1972

  • Topics: Number Theory, Algebra

Publish with us

Policies and ethics

Back to top

Search

Navigation