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A Course in Arithmetic

  • Textbook
  • © 1973

Overview

Authors:
  1. Jean-Pierre Serre

    1. Collège de France, Paris Cedex 05, France

Part of the book series: Graduate Texts in Mathematics (GTM, volume 7)

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About this book

This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). It is achieved in Chapter IV. The first three chapters contain some preliminaries: quadratic reciprocity law, p-adic fields, Hilbert symbols. Chapter V applies the preceding results to integral quadratic forms of discriminant ± I. These forms occur in various questions: modular functions, differential topology, finite groups. The second part (Chapters VI and VII) uses "analytic" methods (holomor­ phic functions). Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no. 2.2). Chapter VII deals with modular forms, and in particular, with theta functions. Some of the quadratic forms of Chapter V reappear here. The two parts correspond to lectures given in 1962 and 1964 to second year students atthe Ecole Normale Superieure. A redaction of these lectures in the form of duplicated notes, was made by J.-J. Sansuc (Chapters I-IV) and J.-P. Ramis and G. Ruget (Chapters VI-VII). They were very useful to me; I extend here my gratitude to their authors.

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-ix
    Download chapter PDF

  2. Algebraic Methods

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. Finite Fields

      • Jean-Pierre Serre
      Pages 3-10

    3. p-Adic Fields

      • Jean-Pierre Serre
      Pages 11-18

    4. Hilbert Symbol

      • Jean-Pierre Serre
      Pages 19-26

    5. Quadratic Forms over Q p and over Q

      • Jean-Pierre Serre
      Pages 27-47

    6. Integral Quadratic Forms with Discriminant ± 1

      • Jean-Pierre Serre
      Pages 48-58

  3. Analytic Methods

    1. Front Matter

      Pages 59-59
      Download chapter PDF

    2. The Theorem on Arithmetic Progressions

      • Jean-Pierre Serre
      Pages 61-76

    3. Modular Forms

      • Jean-Pierre Serre
      Pages 77-111

  4. Back Matter

    Pages 112-119
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Reviews

“The book is a showcase of how some results in classical number theory (the Arithmetic of the title) can be derived quickly using abstract algebra. … There are a reasonable number of worked examples, and they are very well-chosen. … this book will expand your horizons, but you should already have a good knowledge of algebra and of classical number theory before you begin.” (Allen Stenger, MAA Reviews, maa.org, July, 2016)

Authors and Affiliations

  • Collège de France, Paris Cedex 05, France

    Jean-Pierre Serre

Accessibility Information

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Bibliographic Information

  • Book Title: A Course in Arithmetic

  • Authors: Jean-Pierre Serre

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4684-9884-4

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York 1973

  • Hardcover ISBN: 978-0-387-90040-7Published: 29 November 1978

  • eBook ISBN: 978-1-4684-9884-4Published: 06 December 2012

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: IX, 119

  • Additional Information: Original French edition with the title: Cours d'arithmetique

  • Topics: Number Theory

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