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Quadratic Number Fields

  • Textbook
  • © 2021

Overview

Authors:
  1. Franz Lemmermeyer

    1. Jagstzell, Germany

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  • Connects quadratic fields with modern algebraic number theory
  • Applies the theory to solve Diophantine equations
  • Contains hundreds of exercises with solutions
  • Includes original historical commentary

Part of the book series: Springer Undergraduate Mathematics Series (SUMS)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Prehistory

    • Franz Lemmermeyer
    Pages 1-29

  3. Quadratic Number Fields

    • Franz Lemmermeyer
    Pages 31-52

  4. The Modularity Theorem

    • Franz Lemmermeyer
    Pages 53-89

  5. Divisibility in Integral Domains

    • Franz Lemmermeyer
    Pages 91-106

  6. Arithmetic in Some Quadratic Number Fields

    • Franz Lemmermeyer
    Pages 107-134

  7. Ideals in Quadratic Number Fields

    • Franz Lemmermeyer
    Pages 135-163

  8. The Pell Equation

    • Franz Lemmermeyer
    Pages 165-192

  9. Catalan’s Equation

    • Franz Lemmermeyer
    Pages 193-208

  10. Ambiguous Ideal Classes and Quadratic Reciprocity

    • Franz Lemmermeyer
    Pages 209-221

  11. Quadratic Gauss Sums

    • Franz Lemmermeyer
    Pages 223-254

  12. Back Matter

    Pages 255-343
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Keywords

About this book

This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level.

Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study.

Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.

Reviews

“The book is very nicely written and the original style and choices of the topics make it agreeable reading, and might well complement and motivate the study of other classical introductions to the theory of more general number fields.” (Alessandro Cobbe, zbMATH 1498.11003, 2022)

Authors and Affiliations

  • Jagstzell, Germany

    Franz Lemmermeyer

About the author

Franz Lemmermeyer has worked in algebraic number theory and has published several books on the history of number theory, in particular on reciprocity laws.

Bibliographic Information

  • Book Title: Quadratic Number Fields

  • Authors: Franz Lemmermeyer

  • Series Title: Springer Undergraduate Mathematics Series

  • DOI: https://doi.org/10.1007/978-3-030-78652-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Softcover ISBN: 978-3-030-78651-9Published: 19 September 2021

  • eBook ISBN: 978-3-030-78652-6Published: 18 September 2021

  • Series ISSN: 1615-2085

  • Series E-ISSN: 2197-4144

  • Edition Number: 1

  • Number of Pages: XI, 343

  • Number of Illustrations: 8 b/w illustrations, 8 illustrations in colour

  • Topics: Number Theory

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