Galois Theory is, in this reviewer's humble opinion, one of the most beautiful areas of mathematics. Its beauty lies not only in its simple definitions yet deep results, but also in its powerful applications to a wide range of mathematics and its exciting history. David Cox's new book, titled simply Galois Theory, would be a wonderful introduction to this rich area of mathematics. The book is broken into four parts, each of which is further broken into chapters and then sections. One of the best features of the book is that at the end of each section, Cox has written Mathematical Notes — which further develop some of the mathematical ideas that made cameo appearances in the section, or point out where these ideas abut ideas in other areas of mathematics — as well as Historical Notes, which talk about the history of the mathematics, often including excerpts from primary sources by Kronecker, Lagrange, or Galois himself. These notes often contained information which would be new even to an expert in the field, and I think are a wonderful touch when using this book with students, as it can give them places to launch off and learn more on their own.
The first part of the book is about polynomials, developing the theory of the cubic formula as well as discussing symmetric polynomials and other results which will be useful in the development of Galois theory proper. The next part consists of four chapters of fields and field extensions, and also including most of the main results of Galois theory, including the Galois Correspondence. The third part of the book concerns applications of Galois theory, including the standard results about solvability by radicals and geometric constructions (with a nice section on origami) and also a chapter on finite fields and their Galois groups. The final chapters of the book are what Cox describes as "Further Topics", including more on the historical development of the theory, a chapter on computing Galois groups, and more. The book concludes with an appendix which contains the results from Algebra that Cox uses in the main text, which would serve as a nice reference though I cannot imagine that a student would be able to truly learn everything they need to from the appendix.
The writing in the text is of a high quality, and it is littered with examples and explicit computations. Cox also refers to computer packages to help quite a bit (and he gives information for both Maple and Mathematica users!), though one would not need to use these to follow the vast majority of the book. There is a nice bibliography, and a decent number of exercises. In summary, I think that this is a fantastic book, and I would not hesitate to use it in an undergraduate Galois Theory class.
Darren Glass is Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at firstname.lastname@example.org.