算术教程 第1版 A Course in Arithmetic 英文原版 JP Serre

基本信息

Format Hardback | 119 pages

Dimensions 155 x 235 x 14.22mm | 810g

Publication date 10 Oct 1996

Publisher Springer-Verlag New York Inc.

Language English

Edition Statement 1st Corrected ed. 1973. Corr. 3rd printing 1996

Illustrations note IX, 119 p.

ISBN10 0387900403

ISBN13 9780387900407

书籍简介

这本书分为两部分。 第一个是纯代数的。 它的目标是对有理数域上的二次型进行分类(Hasse-Minkowski定理)。 前三章对二次互易定律、p进场、希尔伯特符号进行了初步探讨。 第五章将上述结果应用于判别+/- i的二次积分形式。这种形式出现在各种问题中:模函数、微分拓扑、有限群。 第二部分(第六章和第七章)使用了“解析”方法(全谐函数)。 第六章给出狄利克雷“等差级数定理”的证明; 这个定理在第一部分的一个临界点上被使用。 2.2)。 第七章讨论模形式，特别是函数。 第五章的一些二次形式在这里又出现了。 这两个部分对应于1962年和1964年在高等师范学院给二年级学生的讲座。 j - j以重复笔记的形式对这些讲座进行了修订。 Sansuc(第I-IV章)和j - p。 Ramis和G. Ruget(第七章)。 它们对我非常有用; 在此，我要感谢他们的作者。  

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.

“这本书展示了如何使用抽象代数快速推导出经典数论（标题的算术）中的一些结果。......有合理数量的工作示例，而且它们是非常精心挑选的...... .. 这本书会扩展你的视野，但在开始之前，你应该已经对代数和经典数论有很好的了解。” （Allen Stenger，MAA 评论，maa.org，2016 年 7 月）

"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)

目录

I-Algebraic Methods.- I-Finite fields.- 1-Generalities.- 2-Equations over a finite field.- 3-Quadratic reciprocity law.- Appendix-Another proof of the quadratic reciprocity law.- II - p-adic fields.- 1-The ring Zp and the field Qp.- 2-p-adic equations.- 3-The multiplicative group of Qp.- III-Hilbert symbol.- 1-Local properties.- 2-Global properties.- IV-Quadratic forms over Qp and over Q.- 1-Quadratic forms.- 2-Quadratic forms over Qp.- 3-Quadratic forms over Q.- Appendix-Sums of three squares.- V-Integral quadratic forms with discriminant +/- 1.- 1-Preliminaries.- 2-Statement of results.- 3-Proofs.- II-Analytic Methods.- VI-The theorem on arithmetic progressions.- 1-Characters of finite abelian groups.- 2-Dirichlet series.- 3-Zeta function and L functions.- 4-Density and Dirichlet theorem.- VII-Modular forms.- 1-The modular group.- 2-Modular functions.- 3-The space of modular forms.- 4-Expansions at infinity.- 5-Hecke operators.- 6-Theta functions.- Index of Definitions.- Index of Notations.

作者简介

Jean-Pierre Serre 教授是巴黎法兰西学院著名的法国数学家。

Professor Jean-Pierre Serre ist ein renommierter französischer Mathematiker am College de France in Paris.