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A Course in Arithmetic


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Table of Contents

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.


"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,, July, 2016)

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