Articoli correlati a A Course in Arithmetic: 7
Sinossi
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.
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Recensione
“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)
Contenuti
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.
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