The MAA Guide series — a subset of the Dolciani Mathematical Expositions — is rapidly becoming one of my favorite series of books. I like expository books that provide a quick and interesting entrée into an area of mathematics, or a useful source of examples, and that is precisely what these are. They are also, thanks to careful selection of authors, generally very well-written, informative and particularly useful as a resource for a varied audience. This book, the most recent one in the series (number 8, following books on complex variables, advanced real analysis, real variables, topology, elementary number theory, advanced linear algebra and plane algebraic curves) continues this tradition.
Each volume in this series is addressed to readers who, although mathematically sophisticated, are not experts in the subject matter of the book. The canonical example, I would think, would be graduate students seeking an efficient way of helping prepare for qualifying exams. However, faculty members who haven’t had occasion to work extensively in a given area and who want a quick overview of the basic ideas and how they hang together would also find these books valuable. The emphasis in most of the Guides that I have read (this one most definitely included) is providing a survey of the subject in a reasonably short amount of pages, providing a book that is accessible and informative but likely does not contain the kind of technical detail that, although obviously necessary for complete mastery of the material, may serve as an impediment to a person who just wants to know “what’s what” in an area.
So, for example, this book, like many in the Guide series (one possible exception is Weintraub’s Advanced Linear Algebra) is not really intended as a text. There are no exercises, and most proofs are omitted; some that are fairly easy are provided, though never in the rigid theorem/proof format of most textbooks. Instead of proofs, Gouvêa provides discussions of the results and, quite often, a helpful sort of intuition as to why something should be true. (The author uses the phrase “shadows of proofs” in this connection.) To compensate for the lack of proofs, there is an excellent bibliography, to which the author makes frequent specific references throughout the text.
There are also lots of nice examples. A professional algebraist may be able to immediately give an example of a projective module that is not free, or a ring that does not have the invariance of basis number property, but people who don’t work with algebra all the time may not have such examples on the tip of their tongues. The reader will find such examples here (along with, in connection with the latter, a succinct explanation of why such an example must be noncommutative). The reader will also find some examples that involve completely different branches of mathematics; there is, for example, a nice little one-page discussion of how modular forms arise from group actions, and the author also makes occasional remarks about topics such as topology and elliptic curves. The discussions here are not deep or technical, just brief overviews that give the reader some idea of what the terms mean; perfect for a student or non-specialist faculty member who may wind up hearing the phrase in a talk somewhere. In conformity with the intended readership, examples are not necessarily set off with big margins and the word EXAMPLE in large letters, but are often incorporated directly into the text.
The book is divided into six chapters, the first three of which are largely prefatory to the last three, which in turn comprise the meat of the book. Chapter 1 provides a succinct, interesting historical look at algebra, in which the author briefly tracks the development of algebra from its classical origins through its modern period (i.e., the axiomatic approach of Artin and Noether) up to its “ultramodern” period of category theory. Chapter 2 continues the study of categories; not being a huge fan of what Serge Lang once famously referred to as “abstract nonsense”, I feared, when I saw this early chapter on the subject, that the entire book would be filled with commutative diagrams and exact sequences, but was pleased to discover, as I read on, that Gouvêa does not overdo this; these things generally don’t appear unless their appearance really does enhance the discussion. Chapter 3 is a bestiary of algebraic terms, some of which are re-defined later and discussed in more detail.
The remaining three chapters discuss, in order, the three algebraic structures mentioned in the title of the text: groups, rings and fields (including skew fields). Chapter 4 on groups starts with the definition and then proceeds to discuss all of the general topics that one would expect to encounter in a first year graduate course, and perhaps a somewhat more: the chapter talks about Sylow theory, nilpotence and solvability, the word problem, group representation theory (in characteristic 0) and more. The discussion, even of elementary concepts, is done at a mathematically mature, but nonetheless accessible, level (for example, cosets of a subgroup H of a group G are defined as orbits under a certain group action), which I think is entirely appropriate, given the intended readership, and which also has the advantage of letting the reader see how these ideas really fit into the “big picture” (for example, the fact that distinct cosets partition the group is now seen to be just a special case of the more general result about orbits).
The next chapter is on rings and modules, and here, too, we are treated to an excellent survey of that area of mathematics: basic definitions, followed by discussions of topics such as localization, Weddeburn-Artin theory, the Jacobson radical, factorization theory, Dedekind domains (with a look at algebraic numbers), and various kinds of modules (free, projective, injective, etc.). As in the earlier chapter on group theory, the discussion here is at a mature level, with the author frequently stating things at a somewhat greater level of generality than might usually be encountered. (Examples: a quite general statement of Nakayama’s lemma is given, and the usual results about modules over PIDs are deduced as a special case of the more general situation of modules over Dedekind domains.) Notwithstanding this, however, Gouvêa also keeps the needs of students firmly in mind; for example, there is a section titled “Traps”, in which he points out, with simple specific examples, some of the ways in which modules can differ from vector spaces. (He tells of a friend who once described modules as “vector spaces with traps”.)
The final chapter is on field theory. Galois theory is covered, of course (in a considerably general way, including infinite Galois groups and their topologies) but the chapter also contains material on such topics as algebras over a field, function fields, central simple algebras and the Brauer group.
Because the author is writing for people who already have some mathematical sophistication, including some prior exposure to abstract algebra, he does not feel obliged to follow a strictly linear order of presentation. So, for example, the chapter on groups, which precedes the chapters on rings and fields, nonetheless contains references to things like finite fields, semisimple rings and algebraic numbers; as another example, Nakayama’s Lemma in ring theory is stated in a form involving tensor products, which are not formally discussed until a few sections later. This provides a certain freedom that an author of a strictly introductory text does not have, and helps, I think, enhance one’s overall understanding of the subject by providing a broader point of view than might otherwise be possible. Likewise, even within a chapter, the level of difficulty is not necessarily monotonically increasing, and sometimes fairly sophisticated topics (e.g., profinite groups) are discussed before much more elementary ones (e.g., permutation groups). So, if you find a certain section to be fairly heavy going, just keep reading, and chances are, within a page or two, you will find things more comfortable.
The writing style throughout the book is of uniformly high quality. The author is one of those rare people who has the ability to write like people talk, with a nice, conversational tone that sometimes elicits a smile as well as a nod of understanding. Here, for example, is how he ends his discussion of groups of small order: “The next interesting case is order 16, which is, alas, a bit too interesting. There are five different abelian groups (easy to describe) and there are nine different nonabelian ones (most of them not easy to describe). So we will stop here.” And see also page 160 for a cute little comment that will appeal to fans (of a certain age) that remember Tom Lehrer.
It should be apparent from the preceding discussion that I liked this book — a lot. Nevertheless, it seems inevitable that any reviewer will find some nits to pick, just because no two people will ever write the same book. The ones I have, though, are neither numerous nor particularly significant, and basically just reflect my personal preferences. I would have liked, for example, to have seen an example of non-isomorphic groups with the same character table (Everybody’s Favorite Example is D4 and the quaternion group), as well as a specific example of a rational polynomial of degree 5 that is not solvable (the author states that the “generic” polynomial of degree at least five is not solvable and also states that an irreducible polynomial of prime degree with two real roots and at least one non-real root is not solvable by radicals, but does not give an actual fifth-degree polynomial meeting these conditions). I think the phrase “special linear group” should have been introduced when the group SL(n,K) was first defined on page 33, rather than fifty pages later, and also think that discussing unique factorization without at least mentioning Fermat’s Last Theorem can only be described as a lost opportunity.
Additionally, one of my favorite cute applications of transcendence bases has always been the proof that the field of complex numbers has infinitely many automorphisms (a fact that I think is insufficiently well known); the author develops all the machinery necessary to establish this, but doesn’t say so explicitly. Finally, in connection with the definition of algebraically closed fields, the author states the Fundamental Theorem of Algebra (that the field of complex numbers is algebraically closed) and says that all proofs “depend on the topology of the complex field”. This statement, though true, may lead students to believe that all proofs are very analytic or topological in nature; in fact, there is at least one proof that uses Sylow and Galois theory and only two simple facts from analysis, namely (a) that any real polynomial of odd degree has at least one real root, and (b) that any quadratic polynomial with complex coefficients has a complex root.
But these are quibbles. Overall, this is a valuable book — a pleasure to read, and packed with interesting results. It should be very helpful to graduate students and non-specialists wanting a succinct summary of the subject, and even professional algebraists may find something new and interesting here. It is a splendid addition to an excellent series.
One final comment: in the interest of full disclosure, I should mention that, as faithful readers of this column probably already know, the author of this book is also the editor of this column. This raises, I suppose, at least the question of a conflict of interest. This same issue arose when another of the author’s books, p-adic Numbers, was favorably reviewed in this column by Darren Glass more than two years ago, and since I don’t think that I can improve on the way Professor Glass addressed it, I will simply quote him verbatim: “[T]he reader can rest assured that this reviewer would have said equally flattering things about the book even if it wasn’t written by his editor. Besides, I couldn’t think of anything that an editor could use to bribe his volunteer reviewers with (More prominent placing on the site? First crack at the new Keith Devlin?) so I didn’t even bother asking.”
Mark Hunacek (email@example.com) teaches mathematics at Iowa State University.