It is commonplace to note that there is a similarity between the Galois theory of field extensions and the topological theory of covering spaces and fundamental groups. In both theories there is a “biggest object”: in Galois theory, the algebraic closure (or separable closure) of the base field; in the topological theory, the universal covering space. The automorphisms of this object form a group: the absolute Galois group in one case, the fundamental group in the other. Taking appropriate subgroups, we obtain intermediate objects; in good cases the automorphism groups of these intermediate objects are quotients of the big group by the chosen subgroup.
Galois Groups and Fundamental Groups starts from that observation and sets out to push it as far as possible. It opens with a quick review of classical Galois theory, which is quickly generalized to handle infinite field extensions and restated in the language of category theory and finite étale algebras. A second chapter reviews the theory of fundamental groups and covering spaces. With those in hand, we are ready for the more serious stuff: coverings of Riemann surfaces, fundamental groups of algebraic curves, Grothendieck’s algebraic fundamental group of a scheme, and finally Tannakian categories.
The presentation is clean and efficient. Most of the prospective readers of the book will already know the theories described in the first two or three chapters, which will serve mostly to set up notation, give the basic point of view that the book will take throughout, and perhaps complete their knowledge here and there.
The real meat of the book comes when we move to algebraic curves and to schemes. Alas, the pace does not change at that point, so that things seem to suddenly become quite a lot harder. In a way, reading through the book (say, on the treadmill) is much like the famous colloquium lecture in which everyone understands the first fifteen (or is it five?) minutes but fewer and fewer listeners continue to follow as time passes.
Studying the book carefully, on the other hand, would certainly be an immensely profitable experience. From the beginning, the approach is sophisticated and deep, highlighting the features of each theory that will dovetail nicely with the features of the more advanced theory to follow. Proofs are mostly complete and formal prerequisites are kept to a minimum. To help his serious students, the author has included quite a few exercises, mostly quite difficult (even in the first chapter). There are also many references, which allows one to find more detailed accounts (or more advanced ones). Many exercises come with references as well.
Overall, this is a very nice book that will serve well those who want to learn about the subject and are willing to put in some serious work.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.
Foreword; 1. Galois theory of fields; 2. Fundamental groups in topology; 3. Riemann surfaces; 4. Fundamental groups of algebraic curves; 5. Fundamental groups of schemes; 6. Tannakian fundamental groups; Bibliography; Index.