A Course in Arithmetic

1. Front Matter

   Pages i-ix

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2. Algebraic Methods

   1. Front Matter

      Pages 1-1

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   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

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   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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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 at the 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.

• Arithmetik
• algebra
• arithmetic
• field
• finite field
• finite group
• modular form
• quadratic form
• ring
• zeta function

“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)

• Collège de France, Paris Cedex 05, France

  Jean-Pierre Serre