A Course in Arithmetic / by J-P. Serre.

Format:

Book

Author/Creator:

Serre, Jean-Pierre, 1926- author.

Contributor:

SpringerLink (Online service)

Series:

Graduate texts in mathematics 2197-5612 ; 7.

Graduate Texts in Mathematics, 2197-5612 ; 7

Language:

English

Subjects (All):

Number theory.

Number Theory.

Local Subjects:

Number Theory.

Physical Description:

1 online resource (IX, 118 pages).

Edition:

First edition 1973.

Contained In:

Springer Nature eBook

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1973.

System Details:

text file PDF

Summary:

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, numbers 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.

Contents:

I-Algebraic Methods

I-Finite fields

II - p-adic fields

III-Hilbert symbol

IV-Quadratic forms over Qp and over Q

V-Integral quadratic forms with discriminant ± 1

II-Analytic Methods

VI-The theorem on arithmetic progressions

VII-Modular forms

Index of Definitions

Index of Notations.

Other Format:

Printed edition:

ISBN:

9781468498844

Access Restriction:

Restricted for use by site license.