In a recent interview (Notices of the AMS, June/July 2016, pp. 645–646) Jordan Ellenberg said, “All young number theorists should read Serre’s A Course in Arithmetic; it somehow captures an immense amount of the spirit of the subject in a tiny amount of space.” 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.
For the most part the proofs are not new, but are more insightful ways of looking at old proofs. For example, the proof of quadratic reciprocity is essentially Gauss’s sixth proof from 1818, but expressed in terms of algebraic number fields rather than polynomial identities.
The book assumes you already have a good working knowledge of fields, rings, groups, modules, and linear algebra. It uses everyday results from these fields with no explanation. This produces very short proofs and developments. For example, all the essentials of p-adic numbers are worked out in 8 pages. The proofs are short not because they leave out a lot of steps, but because they assume you already know a lot of background (historically a lot of these number theory topics were raw material for the more abstract algebraic theories).
It’s still not an easy book and requires careful study; the last chapter on modular forms in particular is very intricate. This is not an introductory course in number theory: because of the difficulty, because not much context is given, and because there are no exercises. There are a reasonable number of worked examples, and they are very well-chosen. It’s also not a comprehensive course; it proves a lot of important results, but it is narrowly focused.
The book is divided into two parts. The algebraic part covers mostly quadratic reciprocity and quadratic forms. In general you get your choice of fields for the theorems: either the reals or the p-adic numbers. After the general theory is developed, the book specializes these to get the most important results for quadratic forms with coefficients in the reals, p-adic numbers, rationals, and integers.
The analytic part covers the Dirichlet theorem on primes in arithmetic progressions and modular functions. The two parts are interwoven; the algebraic part uses Dirichlet’s theorem in several places, and the analytic part still has a lot of algebra and builds on some of the theory of quadratic forms for its development of modular functions.
Bottom line: 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 is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. He was once a young number theorist.