In recent decades, the theory of finite fields has become increasingly useful, due to many applications in cryptography and coding theory among other areas. More recently, similar applications have been developed for a class of rings called Galois Rings, which are finite extensions of the cyclic rings Z/pmZ in the same way that finite fields are extensions of cyclic fields Z/pZ. The book under review, Finite Fields and Galois Rings by Zhe-Xian Wan, does not discuss these applications in depth but the motivation of these applications is felt throughout the book.
Wan’s book is based on lectures given in Tianjin and Suzhou and in the preface he writes that an earlier version of the manuscript has been used in courses for students in mathematics, computer science, and engineering. As such, Wan assumes very little in the way of prerequisites, with early chapters in the book introducing the concepts of sets, groups, rings, and fields. The first 100 pages of the book is a crash course in the subject, proving exactly those theorems necessary for the reader to follow the discussions to come about the special cases of finite fields and Galois rings.
The next two chapters are spent on the properties of finite fields, proving the results about their multiplicative group structure, additive group structure, and construction that are familiar to those who have had a graduate course on the topic, as well as topics such as automorphisms, characteristic polynomials, trace and norm. There are then a series of chapters on polynomials over finite fields and how to factor them or figure out when they are irreducible.
After a chapter giving more background on ring theory, Wan discusses Hensel’s Lemma about when one can lift polynomials from Z/pZ to Z/pmZ at length. Finally, he introduces Galois rings as finite rings whose set of zero divisors form a principal ideal. He goes on to prove the structures that Galois rings can take and their relationships with finite fields, as well as discussing their unit groups and automorphisms.
While there are not as many examples as one might hope for from a book that is intended to be used as a textbook, Wan does provide many exercises and references. He is clearly writing for an audience that is somewhat less sophisticated mathematically than many books on these topics, which has all of the advantages and disadvantages that one would expect. The book was somewhat drier and more application driven than I would like, but I imagine that many readers would feel just the opposite. In either event, I think all readers will agree that there is some beautiful and exciting mathematics contained in these pages.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College whose main mathematical interests include number theory, algebraic geometry, and Galois theory. He can be reached at firstname.lastname@example.org.