Algebraic number theory has an unparalleled pedigree. Its prehistory sports none other than Fermat and Euler: just think of Fermat’s Last Theorem and the Law of Quadratic Reciprocity, which Euler knew but never explicitly proved. I believe that a proof can be glued together from results Euler had at his disposal, to be sure, but the first complete proof is credited to Gauss --- who then gave six more. Gauss, with his famous *Disquitiones Arithmeticae*, is likely the one to be credited with giving the greatest impetus to what eventually became this algebraic theory of numbers as a *Ding in Sich*; certainly, the subject, while still not autonomously defined, took off spectacularly in his wake. Indeed we encounter a host of players throughout the later 19th century, including Eisenstein, Sophie Germain, Dedekind, Dirichlet, Kummer, Kronecker, and so on: I am sure I am missing some obvious others. However, it is undeniable that by the end of that century, this discipline was ready for a proper definition, and it was Hilbert’s famous *Zahlbericht* that did the job.

Of course, Hilbert did a great deal more than collect and present the known number theoretic material surrounding algebraic numbers, i.e. roots of polynomials with integer coefficients. For one thing, he gave a very far-reaching extension of Kronecker’s work leading to modern class field theory. For another, Hilbert gave the definitive treatment of reciprocity laws, going well beyond the quadratic one (Gauss-Euler), covering all degrees and all algebraic base fields. So the stage was set: in proper order algebraic number theory could be taught as a well-defined university course, and texts began to emerge. These included what André Weil characterized in his *Basic Number Theory* as the premier source for this material given in classical language, namely, *Vorlesungen über die Theorie der algebraischen Zahlkörper*, or *Lectures on the Theory of Algebraic Number Fields*, by Hilbert’s student Erich Hecke. In any case, the game was afoot, and many authors began to contribute their own versions of the subject, including Hermann Weyl, Emil Artin, and Helmut Hasse. As the 20th century marched on and research on this and allied parts of number theory intensified and expanded, other books appeared. Many of these are now considered standards, e.g. the aforementioned book by Weil, in which idèles and adèles are featured with gusto; the pedagogical masterpiece, *Number Theory*, by Borevich and Shafarevich; and of course the Brighton Conference Proceedings, a.k.a. *Algebraic Number Theory*, by Cassels and Fröhlich. Each and every one of these books is in its own way fabulous and vital for the number theorist to learn his craft, and it is in this collection that we should fit the book under review.

What most distinguishes the many books by Serge Lang is their specific focus on teaching the indicated subject to the prepared student. In this aspect, they are probably unsurpassed as excellent sources for serious courses in a modern doctoral program: Lang intended them for specifically that purpose, and this is certainly the case for *Algebraic Number Theory.* In contrast with, e.g., Borevich-Shafarevich, Cassels-Fröhlich, and Weil, Lang occupies a very different place. Borevich-Shafarevich is a masterpiece, but it cuts a much broader swath than Lang’s book, and requires the reader to develop other themes, whose roles are initially unclear to the novice. The reader should have some experience with algebraic number theory already before he opens this book. Cassels-Fröhlich is encyclopedic, terse and authoritative; every number theorist should have a copy (and read it, study it, and annotate it), but it’s not the place to start --- Lang’s book, on the other hand, is. And this contrast is even more stark when we take a look at another masterpiece, Weil’s *Basic Number Theory*. In fact, this truly marvelous book, by a true grandmaster of the theory of numbers, simply should not be touched unless the reader has already served a serious apprenticeship. I love Weil’s work, but it would be terribly disingenuous to suggest that it is accessible to any but the seriously initiated. *Basic Number Theory* is basic only in that what Weil develops in its pages is vital for modern research in algebraic number theory; it is also a beautifully written book, elegant and concise in the manner associated with Weil --- but it is certainly not for the beginner. That beginner would do well to start with Lang’s book.

All right, then, how does one define “beginner” in this context? What should a rookie aspiring number theorist know before going at Lang’s book, or taking a graduate course that uses this book? I would say that the standard first-year graduate course in algebra (groups, rings, fields, vector spaces: Lang’s Algebra is an excellent source) is non-negotiable, and it would be beneficial if said rookie had some exposure to commutative algebra (say, via Atiyah-MacDonald, *Introduction to Commutative Algebra*). Otherwise, it’s probably optimal to get one’s first exposure to the beautiful and standard fare of the subject straight from Lang’s very well-crafted book.

Caveat: to Lang’s credit, and as a testimony to the fact that Lang was properly influenced by developments in number theory that were set into motion by such mathematicians as Artin, Tate, and Weil, the book under review covers, in part one, idèles and adèles. These are very important topics, indeed, but are a bit austere; to get at these, it would be useful if the reader were to have a pretty solid grounding in point set topology, if only at the undergraduate level.

It is obvious that Lang’

*Algebraic Number Theory* is a very well written and important book: a major player in the sweepstakes for which book to use to learn this sub-discipline well --- very well. It comes very highly recommended.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.