This is an AMS/Chelsea reprint of a book first published in 1967 by Gordon and Breach, based on a series of lectures given at Princeton in 1950–51. The course was a revised version of one offered at New York University in the summer of 1950, the notes for which were published in 1951 by NYU. So what we have here is a record of how Emil Artin presented algebraic number theory and its close cognate, the theory of algebraic function fields, in the early 1950s.

Artin was, of course, a master of the subject. The analogy between number fields and function fields (in one variable) was an important theme in his work, from his thesis onward. Here, he takes a unified approach to both kinds of fields, based on the theory of valuations and of complete valued fields. Artin seems to have taken this approach to the subject already in 1933 (see Artin, "Algebraische Zahlentheorie," Hamburger Beiträge zur Geschichte der Mathematik. *Mitt. Math. Ges. Hamburg* **21** (2002), 159–223).

The first two parts are strictly local, and cover the theory of extensions of complete valued fields (Part I) and local class field theory (Part II). They stike me as fairly "standard," probably because Artin's approach was so influential that it was adopted almost universally.

The third chapter is more interesting. It introduces the notion of "PF-Fields," i.e., fields in which the product formula is true. These turn out, of course, to be algebraic number fields and algebraic function fields in one variable. Tensor products of fields (here called "Kronecker products") are used to treat the semi-local theory and "valuation vectors" (now called adèles) and idèles are used to move from local to global results. Part III culminates with a treatment of differentials and the Riemann-Roch theorem.

What is obviously missing here is a treatment of global class field theory. This was left to a second set of lecture notes, which became, I think, the well-known *Class Field Theory*, by Artin and Tate.

The exposition is (as usual with Artin) quite elegant, and the parallel treatment of number fields and function fields can be illuminating as well as efficient. But this remains a set of lecture notes, so the reader shouldn't expect much discussion and contextualization. An example is the chapter on the Riemann-Roch theorem, in which two whole sections of preparation (one of which is entitled "The First Proof") go by before the theorem is even stated. Similarly, while the notion of a differential is defined for all PF-fields, there is little discussion of how this relates to the classical notion in the function field case, nor of what it means in the number field case.

Such problems, however, are only to be expected in a book like this. They are more than compensated for by the insight that will come from working through it, particularly if the reader is prepared to sort out how this approach relates to more classical (or more modern!) ones. Thus, students of number theory and algebraic geometry can learn a lot from this book. It is a true classic in the field.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. He first learned this material from J.-P. Serre's *Local Fields* and André Weil's *Basic NumberTheory*, and he is delighted to add Artin's book to the same shelf as those two other classics*.*