This book, a translation from a French text published in 2012 that arose from the authors’ experiences teaching this material at the Ecole Polytechnique in France, is an interesting and useful one, but one that is, I think, in the wrong Springer series. Most or all of the other volumes in the Springer Undergraduate Mathematics Series (SUMS) are, as the name implies, suitable as texts for undergraduate courses in typical American universities. This one is not. 

The subject matter of this book, Galois theory, is one that, unfortunately, quite a few American undergraduate mathematics majors never get to see. Those that do get to see it usually do so at the end of a second semester of abstract algebra, and only spend a week or two on it, learning the very basics. For that reason, a book that makes this material accessible to undergraduates at a deeper level (as do, for example, the books on this subject by Ian Stewart or David Cox) is a valuable addition to the literature. But it is a rare American undergraduate that will be able to read this book with profit. For one thing, it is very terse: there are only 141 pages of text, followed by roughly 40 pages of problems (to complement the ones in the earlier chapters of the text) and solutions. The authors’ concise writing style requires a reader to have a degree of mathematical maturity that, these days, may be beyond most American undergraduates.

The book begins with a quick look at two classical problems (ruler and compass constructions and solution of polynomial equations) and an explanation of how Galois theory is involved in their solution. Chapters 2 and 3 review facts about group and ring theory, respectively, starting with the definitions of these objects. (The authors’ “ring” is what I grew up calling a commutative ring with identity.) Starting from scratch, the authors cover, among other topics: exact sequences, group actions, symmetric groups, solvable groups, ideals and quotient rings, polynomial rings, modules, the Chinese Remainder Theorem (called here the “Chinese lemma”), and the Frobenius automorphism. (The discussion of Frobenius automorphisms leads to the unfortunate statement that a ring of prime characteristic “always has a non-trivial endomorphism”.)

Chapter 4 discusses algebras over a field. Because the main example here is a field viewed as an algebra over a subfield, algebraic and transcendental elements over a field play a prominent role in this chapter. The quotient ring construction of a field extension of a field F containing a root of a polynomial over F is given, the authors using the phrase “rupture field” (which was new to me) to refer to this larger field.

Chapter 5 discusses finite fields. An unusual feature here is a discussion of the Berlekamp algorithm for factoring polynomials over such fields. The concept of a perfect field is introduced, and it is proved that every finite field is perfect. By working with perfect fields, the authors are able to avoid discussing general separable extensions.

Galois theory, per se, commences in chapter 6. In the space of ten pages, the authors define Galois extensions and prove the Fundamental Theorem of Galois Theory. This is followed by a chapter which introduces infinite Galois extensions. (This theory involves putting a topology on the Galois groups, called the Krull topology or, here, the profinite topology; the authors assume for this chapter that the reader is familiar with the rudiments of point-set topology.)

In chapter 8, which circles back to chapter 1, the authors discuss cyclotomic extensions and cyclotomic polynomials, and apply these ideas to prove a necessary and sufficient condition for a regular n-gon to be constructible.  (It seems a pity that, having addressed the question of what points in the complex plane are constructible, the authors do not also discuss other construction problems, such as angle trisection.)  Chapter 9 then addresses the second of the two classical problems discussed in chapter 1, and proves that the general fifth degree polynomial is not solvable by radicals. 

Chapter 10 is an interesting look at the method of “reduction mod p” for studying the Galois groups of separable polynomials with integer coefficients. This seems to be material that is not often covered in books at this level.

To summarize and conclude: though I cannot recommend this book for undergraduate use here in the United States, it is a viable candidate for a graduate course in Galois theory. Even if not adopted, the numerous problems and solutions might be a useful resource for faculty.

Mark Hunacek (mhunacek@iastate.edu) is a Teaching Professor Emeritus at Iowa State University.