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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)
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Front Matter
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Algebraic Methods
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Front Matter
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Analytic Methods
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Front Matter
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Back Matter
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)
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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
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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