Virtually everyone who has taken or taught an undergraduate abstract algebra course knows the order of topics is groups, then rings, then fields. But have you ever thought about why we do the topics in this order? Is it because the list of axioms for groups is the shortest, and the list of field axioms is the longest? Surely something with fewer axioms must be easier to understand, right? This book challenges this conventional thinking. The motivation for doing so is the premise that rings are inherently easier to understand than groups, and that examples of rings familiar to students are quite plentiful. As such this book begins with an extensive study of rings, then discusses groups, and finally fields. While I was quite skeptical of this approach at first (having been taught these topics in the "standard" order), I was quickly won over by the book for this and many other reasons.

The book is very complete, containing more than enough material for a two semester course in undergraduate abstract algebra (and weighing in at over 600 pages). Even though there was a great deal of material presented, I found the book to be very well organized. It is divided into large Sections by major topic, such as "Ring Homomorphisms and Ideals," "Groups," and "Vector Spaces and Field Extensions." Each Section is divided into Chapters by main concepts (typically 15 to 20 pages), which are then further subdivided by specific topic. This system made it easy to locate any subject in the table of contents.

The book begins (as most such books do) with a Chapter on preliminaries covering basic properties of the natural numbers, mathematical induction, well ordering, and the axiomatic method. I like the exercises at the end of this Chapter, particularly those on induction, because they provide the students with some interesting contexts (the triangle inequality, complete graphs, the binomial theorem) in which to apply the ideas discussed. The remaining Chapters in this Section discuss in depth the properties of the integers, then the integers modulo n, and finally polynomials with rational coefficients (presented in order of increasing abstractness), without mention of the word ring or the ring axioms. Once the students feel comfortable with these examples, the general notion of a ring is introduced in the next Section and the book takes off from here.

I agree with the authors' premise that rings are a better place to start in a first abstract algebra course than groups. Most of the examples of groups that we give students are also rings, and it can be confusing to the students to remember which operation they using to form a group. Additionally, I believe students benefit from a little more mathematical maturity before trying to understand standard examples of non-abelian groups (dihedral and symmetric groups, for instance). Contrast these with standard examples of rings: the integers, the integers modulo n, spaces of polynomials over a field, spaces of functions over a field, and matrices over a field. These are all ideas with which most math majors will have worked a great deal; calling them "rings" just puts familiar objects in a different light.

One of my only negative comments about the book is that it covers a great deal of ring theory (homomorphisms, ideals, integral domains, factorization and UFDs, PIDs, Euclidean domains, maximal and prime ideals, the Chinese Remainder Theorem) before even mentioning the word "group." An instructor following this text, it seems, would need to skip around in order to fit groups into the first semester of an abstract algebra sequence. While I agree that groups are more difficult to understand for most students, I still think they are an important concept which belongs in a first semester course.

There are a lot of things that I like about this book. At the end of each Chapter there is a Chapter Summary, and a Section Summary at the end of each Section. I really think these are well written and will help students to see the big picture. The book definitely seems to be written for students instead of instructors. It gives the motivation behind each discussion and goes to great lengths to explain each idea and theorem in detail — the chapters on constructibility seemed particularly well done to me. The exercises at the end of each Chapter are well written and thoughtful. There are also little exercises sprinkled throughout the text to aid in student understanding of statements and proofs (things which a mathematician would know to work out for himself, but an undergraduate student would not). All in all it seems that a lot of thought went into this book, resulting in a comprehensive, well-written, readable book for undergraduates first learning abstract algebra.

Frederick M. Butler is Assistant Professor of Mathematics at the Institute for Mathematics Learning, West Virginia University.