A Course in Arithmetic

by Jean-Pierre Serre

Book Overview

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:...

• Format: Hardcover
• Language: English
• ISBN: 0387900403
• ISBN13: 9780387900407
• Release Date: November 1978
• Publisher: Springer
• Length: 119 Pages
• Weight: 0.80 lbs.
• Dimensions: 0.6" x 6.3" x 9.6"

Customer Reviews

An excellent book, and a great title!

Contrary to what many Americans may think of when they hear "arithmetic", this book is not about adding and multiplying (well, at least not at a basic level). Rather, this book covers several important topics in number theory - that of quadratic forms (the "algebraic" portion of the book), L-functions associated to Dirichlet characters, and modular forms (the "analytic" portion). The section on quadratic forms is primarily devoted to a proof of the Hasse-Minkowski theorem, which is a type of "local-global" principle. This provides a natural opportunity to introduce p-adic numbers, which have become ubiquitous in the algebraic side of number theory. The analytic section starts with basic definitions and provides a proof of Dirichlet's theorem on primes in an arithmetic progression, which provides a natural opportunity to introduce an initial theory of L-functions. The last section provides an excellent introduction to the basic properties and theorems on modular forms. What's the common theme in this book? We have the opportunity to see the origins of essential ideas in modern number theory (local-global principle, modular forms, L-functions) in a natural way. On the one hand, this book (as the other reviewers have mentioned) can be quite difficult for the beginner to read, since the exposition is sparse (but superb) and the material quite non-trivial. On the other hand, modern number theory (and much of modern mathematics) is incredibly abstract and advanced, and to see these ideas presented by a master is perhaps the best way to be introduced to the modern practice of number theory. Ultimately, with respect to difficulty it doesn't matter who or what you read to learn number theory. The subject will always be daunting to all but the most talented students, but any effort you put into reading this book is well worth it. In summary, any aspiring number theorist (or person interested in modern number theory) needs to read this book. It can serve as an introductory text (I read it as such), but it is also an excellent reference, both for the mathematical material inside as well as the outstanding mathematical exposition. It's rare nowadays to see a mathematician of Serre's caliber also excel as an expository writer/teacher, so this book is a true gem.

not to be confused with an "arithmetic" book

In french, "arithmétique" means elementary study of rational numbers... You will not learn how to add or substract rational numbers here. An easy book to begin with this kind of maths is Andrew's: number theory (from Dover): well explained with a lot of examples...I guess that the author meant something like: "arithmetic of quadratic forms..." The truth is contained in the book's introduction: the text is made of lessons given to second-year "normaliens" from the "rue d'Ulm": those were (and still are) among the most gifted mathematic students in France; (post-graduate level...) therefore any reader without a really sharp mind and a very good level of knowledge in algebra will hardly grasp anything from the book which is really demanding; in the first ten pages, you will find a study of finite fields aimed at proving the celebrated "law of quadratic reciprocity"; it is already well beyond classical elementary arithmetic as it was taught in "classe terminale" thirty years ago and furthermore nowdays. Then it tackles with p-adic fields, entire quadratic forms and then Dirichlet Theorem on primes in arithmetic sequences in a mere 20 pages (to set things up, this theorem is not proven in Hardy an Wright's: "introduction to the theory of numbers" because it is not an elementary theorem as mathematicians go since any proof of it requires use of advanced complex variable methods...) and at last modular forms; as you can expect, everything in the book is connected; so it looks like (it is...) an impressive piece of mathematics; it is a daunting but not impossible task to go through the whole book; let say that such hard work is rewarding and the chapter on modular forms is really a fascinating one ( and a first step towards Wiles' proof of Fermat's last theorem albeit so many more steps have to be climbed to achieve that precise aim...) ; moreover, the chapters using "analytic methods" can be read independantly; "un tour de force".

Breathtaking

Serre's work could best be summarized in one word - Elegance.The book comprises of two distinct parts. The first one is the 'algebraic' part. Serre's goal in this section is to give a complete classification of the quadratic forms over the rationals. As preliminaries to reaching this goal, he introduces the reader to quadratic reciprocity, p-adic fields and the Hilbert Symbol. After these three, he spends the next chapter detailing the properties of quadratic forms over Q and Q_p (the p-adic field). The reason to work over Q_p is the Hasse-Minkowski Theorem (which says that if you have a quadratic form, it has solutions in Q if and only if it has solutions in Q_p). Using Hensels Lemma, checking for solutions in Q_p is (almost) as easy as checking for solutions in Z/pZ. After doing that, he spends yet another chapter talking about the quadratic forms over the integers. (Note: the classification goal is already achieved in previous chapter).The second half of the book is the 'analytic' one. The first chapter in this section gives a complete proof of Dirichlet's theorem while the second one studies the properties of modular forms (these are good!)Due to the extreme elegance, the book is sometimes hard to read. This might sound like a paradox, but it's not and I'll explain why. The book takes some effort to read because it's terse and it often takes a while to figure out why something is 'obvious'. However, once you see it all, you'll realize that a great mind was guiding you through the pursuit. The choice of topics is just right to achieve the goals that the author sets out for himself. Also, I'd rather think for myself and read a smaller book than be given a huge fat tome where the author details his own thought process.This book was my first foray into number theory and I absolutely enjoyed it. If you're considering reading it, I wish you joy in your pursuits.

Very Demanding

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.

A sheer delight

This short book on number theory by one of the giants of 20th century mathematics is delightful to read. Its length motivates one to finish the book, and it is packed full of interesting results. Most of the theory discussed in the book has wide-ranging applications, such as cryptography and dynamical systems. The last chapter of the book is the best and covers the subject of modular forms, including theta functions, Hecke operators, and general modular functions. If you want to understand the Wiles proof of Fermat's last theorem, start with this book.