The importance and central role of number theory in mathematics might be the explanation for the fact that, of the about fifty Fields medalists since 1950, at least ten have made important contributions to number theory. The author of the book under review is one of those Fields medalists. His book, as its title indicates, introduces many branches of number theory. After writing, many years ago, a Concise Introduction to the subject, this time the author has set out to describe many important concepts and results in number theory in, as the title says, a comprehensive form. The resulting book is suitable for a wide range of audiences.

The initial chapters give a quick review of fundamental results such as divisibility, prime numbers, and congruence. The author then enters into algebraic topics and gives an introduction to number fields and their basic theory. Moving to the analytic side of the theory, he describes the foundations of the theory, introducing some of its main heroes and their work, including the distribution of prime numbers among natural numbers and in a given arithmetical progression, sieve methods, and the circle method. The final chapter is an introduction to elliptic curves. Each chapter ends with comments and references for further study, followed by exercises.

The book is written in a classical style. The proofs are at the heart of the author’s description of the concepts. There is no numbering for formulae throughout the book and only few distinguished results are labeled as theorems and lemmas. This might make the book a little hard to use in a standard course, but certainly it makes the book very friendly for personal use and study.

Anyone who would like to know what is happening in number theory will find the book under review helpful. It paints a clear picture of the foundations and the important directions in number theory. I strongly recommend this book for students at the graduate and research level wishing to improve their global knowledge of number theory.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His field of interest is Elementary, Analytic and Probabilistic Number Theory.