Algebraic Theory of Numbers

I have heard it said that it is impossible to learn algebraic number theory from just one book. There are so many points of view, so many different technical approaches, it is said, that one has to be exposed to several different accounts of the theory in order to see it whole.

That may well be the case, but one needs to start somewhere. The standard recommendation among number theorists has been Number Fields, by Daniel A. Marcus. With this new Dover edition, Pierre Samuel’s Algebraic Theory of Numbers becomes a serious contender for the title of best introduction to the field — at least for certain readers.

Perhaps the first thing to say is that this is a very French book. What I mean is that it exemplifies a style that was typical of French mathematical writing in second half of the twentieth century: brief, precise, and to the point, focusing more on clean proofs of the central theorems than on examples. Even the book’s layout fits the prototype: very long lines of closely-spaced type, theorems and other theorem-like structures in italics, the same font as in many French books with silver covers published by Hermann.

The aim is to be elegant and clear. For a book like this to work, one needs an author who has a deep understanding of the mathematics and the ability to explain it well. Pierre Samuel is just such an author. He knows exactly where he wants to go and has a knack for focusing on the core idea of a proof. He assumes that his readers have a working knowledge of the fundamentals of algebra, and uses this material fearlessly. I have always admired his two-page account (in chapter one!) of the main structure theorem for modules of finite type over principal ideal domains, complete with the corollary describing all finite abelian groups.

Samuel limits himself to the basics: rings of integers in number fields, unique factorization of ideals, how prime ideals factor in field extensions, the finiteness of the ideal class group, the structure theorem for the group of units, and how all this interacts with Galois theory. There are no L-functions, no valuation theory, no local fields, no adèles, no cohomology, and most certainly no class field theory. For all that, the reader is referred to the books in the lightly-annotated bibliography at the end.

The book's virtues are, of course, also its faults. Readers who need TLC from their authors should go elsewhere, as should those who have forgotten some of their linear algebra or are uncomfortable with quotient rings and other abstract constructions. Those interested in algorithmic or analytic questions will also not find what they need here. Finally, the exercises are quite hard.

All of this means that this book will work best for a certain kind of student: comfortable with abstraction, motivated, willing to put in the effort to decode and expand, able to find pleasure in working through the exercises. They will find here a fantastic way to get started on the theory of algebraic numbers.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.