I am afraid there aren’t all that many textbooks that cover both the theory of algebraic number fields and that of algebraic function fields: plenty that deal with the first --- the second? Not so much. The first one that comes to mind for me is the classic book by Emil Artin, *Algebraic Numbers and Algebraic Functions*. I studied that book rather closely many years ago, right after doing the same to what, to my mind, is still one of the very best sources on algebraic number theory proper, Erich Hecke’s *Vorlesungen über die Theorie der Algebraischen Zahlen*. Happily, I am joined in this perhaps idiosyncratic opinion by none other than André Weil: I think one finds this appraisal in his (unquestionably idiosyncratic) book, *Basic Number Theory*. À propos, I guess what this attests to is the sweep of modern number theory: Weil’s book is characterized by its expansive use of idèles and adèles. This perspective was introduced into number theory in John Tate’s famous thesis, "Fourier analysis in number fields, and Hecke's zeta-functions." This title is very suggestive in that there is a direct reference to Hecke whose perspective is really all but non-negotiable, wedding as it does analytic perspectives to algebraic ones; this certainly marks his *Vorlesungen*, just mentioned. Tate’s thesis was written in 1950, under the direction of, yes, Emil Artin, and so in a way it’s all in the family.

By the way, in addition to Weil’s book and Tate’s thesis, the fledgling number theorist keen on learning the ins and outs of the powerful topology afforded by the adèles and (multiplicatively) the idèles, is not restricted to the aforementioned sources. There is also Cassels and Fröhlich, *Algebraic Number Theory*, a.k.a. the 1965 Brighton Conference Proceedings, which is the definitive source on this material. It properly includes Tate’s thesis as a chapter. It may be the prime contender for the title of most important book for modern algebraic number theory: it has it all.

Well, fine, what about algebraic functions, then, i.e. algebraic function fields? Well, André Weil regarded these as the sort of thing that should be treated on an equal footing (or almost so) with number fields, and that’s how he did it in *Basic Number Theory*. But that book, much though it is a masterpiece, is on the austere side, shall we say: it repays doing battle with it, but it is heavy going. Artin’s book is unquestionably more accessible, and so, certainly, is the book under review, Halter-Koch’s *Invitation* to this subject (or subjects). Halter-Koch in fact credits Artin’s book as one of the primary inspirations for his book; he also mentions Weil, noting its elegance and concision, as well as its ambition. So be it. By the way, as he sets about these observations in his Preface, Halter-Koch also cites another classic book I have some fondness for, namely, Helmut Hasse’s book, *Number Theory*, a gem and then some: see, e.g., Irving Kaplansky’s 1981 BAMS review. This fat book is truly encyclopedic and serves as a reference *sine qua non* for this material, but it is difficult to read, perhaps due to Hasse’s notation. It is, as Halter-Koch puts it, easier to access than Weil’s book; I do prefer to wrestle with Weil, however.

All this having been said, Halter-Koch’s book does share a great deal with all the books just mentioned: the approach is valuation theoretic, and the raisons d’être of his book include setting the stage for class field theory. Of course, this raises another point. Hasse and Weil include class field theory in the aforementioned monographs, whereas Artin certainly does not. In point of fact, the definitive (and original) cohomological treatment of class field theory is given by Artin and Tate in the book by that title; here one should also mention the gorgeous book by Jürgen Neukirch, again carrying the indicated title. And so Halter-Koch’s book also eschews this subject: he notes that “the material turned out to be too comprehensive to be published in one volume.” Thus, his Invitation serves to lay the foundations for his “forthcoming volume on class field theory [which] will contain the main theorems for class field theory for local and global fields with the necessary additional foundations from topology, homological algebra, and the theory of simple algebras.” So, we’re dealing with pretty much what Cassels-Fröhlich presents, with material from Hasse added. But Halter-Koch aims to present the entire sweep of this swath of number theory in a coherent way, through specifically targeted pedagogical development. Specifically, his Leitfaden, as given by his chapters, goes like this: algebraic preliminaries (Lang, Jacobson, Hungerford, so sagt der Halter-Koch); ring theoretic foundations (number fields plus function fields, of course) including the needed ideal theory (here I need to mention Serre’s Corps Locaux); an intermezzo glimpse of proto class field theory (especially regarding quadratic extensions); analytic results (my favorite stuff: somewhere Dirichlet is smiling); and then, in the last two chapters, heavy coverage of valuation theoretic stuff and function fields proper (including Riemann-Roch and Hasse-Weil: that tells you the reach of the treatment --- serious stuff indeed!). And there’s the lay-out.

Regarding the nuts-and-bolts of this book: well, it’s excellent. Halter-Koch is extremely thorough, very incisive, and very careful --- all great pedagogical virtues, present in spades. He arranges his results very well, phrasing things carefully and explicitly, and his proofs are detailed. I tend to cover the margins of the books I read with everything from disputes and questions to proof sketches. Halter-Koch’s book would require only a minimum of this sort of polemics: it’s all there --- no guesswork. The additional blood, sweat, and tears attending learning mathematics well, i.e. doing problems, problems, problems, is represented by 20 problems attached to each of Halter-Koch’s six chapters. Scanning them, they look excellent to me: they should serve the reader very well indeed. And that’s true for the entire book: it’s excellent and is well worth using in order to learn this beautiful material. I look forward to Halter-Koch’s book of class field theory!

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