A Course in Arithmetic (Graduate Texts in Mathematics, Vol. 7) (Graduate Texts in Mathematics, 7) by J-P. Serre (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review “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)

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Serre is a great writer. He can be terse, and thus reading him is demanding, but he is careful and is not terse out of laziness, so reading him is possible. You won’t invest time reading a chapter and then find mistakes or omissions or get hopelessly confused and then stop reading the book in frustration. I think this book would be excellent for an undergraduate reading course; probably the book is too long for one semester, but it is nicely divided into two parts, algebraic and analytic, that can be read separately. I like the chapter on p-adic fields; to digest it one needs to be familiar with the ideas of projective systems and exact sequences. The chapter on modular forms is a self-contained introduction to the topic. However, as an introduction to modular forms I prefer Bump’s introduction, in the first chapter of his “Automorphic Forms and Representations”, and when I first learned about modular forms I used both Serre and Bump.

⭐The book is divided into two parts — algebraic and analytic. I’ve only worked through the analytic part. Anything by Serre is worth its weight in gold and this book is no exception; everything Serre covers is of the utmost importance. But Serre’s style is extremely condensed and spare, and he makes no concessions to the reader in terms of motivation or examples. I can’t digest more than half a page of Serre a day; however if one wants to understand the structure of a theory, Serre is ideal.I worked through “A Course in Arithmetic” over a decade back. As I recall I covered Riemann’s zeta function and the Prime Number Theorem, the proof of Dirichlet’s theorem on primes in arithmetical progressions using group characters in the context of arithmetical functions, and some of the basic theory of modular functions. All of this material is also covered in Apostol’s two books on analytic number theory (“Introduction to Analytic Number Theory”, and “Dirichlet Series and Modular Functions in Number Theory”); Apostol goes further than Serre in the analytic part — which is only to be expected since he is devoting two whole texts to the subject.

⭐I recently bought a copy on Amazon of this (terse but excellent) classic book. On the copyright page it says “Printed and bound by R. R. Donnelley and Sons, Harrisonburg, VA.” The print quality is noticeably poor — the text is gray, and the detail is coarse, like an old low-resolution laser printer. There are also a few little blips where, for example, part of a letter will fail to print. At roughly $1 per two-sided page, you would expect better. I don’t know if all copies look like this, in all other respects it appears to be a normal GTM. I wonder if Springer might be using some kind of print-on-demand service for older, lower volume titles that might otherwise fall out of print.

⭐J.P. Serre, the author, is one of the greatest mathematicians of the past half century.He is also a renowned expositor of advanced mathematics. The book inquestion is a valuable reference in a very active area of modern mathematics.PK

⭐A classic. Also fun to keep around to hear people say “I thought you had a PhD. Shouldn’t you know arithmetic?”

⭐Here I am trying to help out some of the more clever upcoming 3rd graders. Their parents contacted me asking about some math books their kids could do over the summer to keep them sharp and ready for next year.I said get a book on Arithmetic, and looked up this one on Amazon and sent them a link to buy it.Well long story short, the parents are coming unglued, griping about common core and how confusing all this stuff is and how they never had to do this when they were kids [what a bunch of “math is hard” whiners]. They claim they have no idea what the book is about, and their little genius snowflakes are crying because they can’t go to Disneyland until they finish this book. I mean for Pete’s sake they should have had this stuff memorized by the end of 2nd grade.Need to go talk to my 2nd grade colleague and found what she was doing all day (I thought I smelled liquor on her breath) since these supposed “smart kids” are dumber than a box of rocks if they can’t handle arithmetic…geesh parents these days.

⭐wow.

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