Foundations of Galois Theory

One of the difficulties in introducing undergraduates to abstract algebra is justifying the abstract approach. The best way to do that is to show them a truly spectacular application. By general consensus, the best choice is Galois theory, leading to the famous theorems about solvability and unsolvability of polynomial equations. It's a good choice, because Galois theory is fairly easy to get to within an undergraduate course. Polynomials are fairly familiar entities, and solving equations is something our students have been doing for a long time.

Books on Galois theory seem to fall into two classes: very short and quite long. The authors of very short books make a beeline for the main theorems, prove them in the most efficient manner (and usually only in characteristic zero), and lay down their pens. Longer books have several choices: spend a lot of time on preliminary material, explore the historical background, develop the theory in a more general setting, or go into more advanced material.

Postnikov's little book is in the same family as the books by Artin and Rotman: short and to the point. First, the way is made simpler by accepting several things as given: for example, that the complex numbers exist and form an algebraically closed field. The argument is restricted to characteristic zero, and in fact to subfields of C , though the author does point out (and make use of, towards the end) the fact that everything goes through for more general fields of characteristic zero. The author takes many facts about polynomials as known; for example, he assumes the reader knows that irreducible polynomials (in characteristic zero) cannot have double roots. He also assumes (and uses) the theory of symmetric polynomials in several variables.

The argument is brisk and efficient. There are no examples; "it is assumed that these will be provided in the course of lectures or seminars," the author says. In order to avoid heavier algebraic prerequisites, Postnikov does not prove or use the theorem about extensions of field homomorphisms. (This makes his approach interesting to those who know the more usual way of proving the main theorem; it's not clear to me that it actually makes the proofs simpler.) Most of elementary group theory is developed, though these sections may not have enough detail for students who have never seen this material before.

The final chapters apply the theory to the solution of equations. In order to consider the general quintic, Postnikov introduces function fields. Here we see another quirky deviation from the standard approach: rather than proving the existence of algebraic closures in general, Postnikov gives an explicit construction of the algebraic closure of a function field (under the assumption that C is algebraically closed). He concludes the final chapter not only with a proof of the unsolvability of the general quintic, but also with the explanation of how the theory leads to the classical formulas for solving the quadratic and cubic equations. (The quartic, alas, is "left to the reader.")

Overall, this is a beautiful little book. All of the basic theorems (in characteristic zero) are here, done clearly, creatively, and efficiently. It will certainly help anyone who is teaching the subject. I suspect that many students will find it too efficient and brisk, and will prefer a text with more examples and more context. Used as it is intended, however, as a supplement to a lecture course, it may just do the trick.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College and editor of MAA Reviews.