This is a text for a one-year course in abstract algebra at the first-year graduate level. Not surprisingly, therefore, it has a lot of textual content in common with other books in this market (*Abstract Algebra* by Dummit and Foote, *Algebra* by Lang, *Advanced Modern Algebra* by Rotman, *Algebra* by Hungerford, *Basic Algebra (I *and* II)* by Jacobson, and *Algebra: A Graduate Course* by Isaacs, to name a few that come readily to mind), but it also has some distinguishing features that, depending on your viewpoint and predilections, may or may not be a good thing.

First, the areas of commonality: like all the books mentioned above, this one concentrates on groups, rings and fields. With regard to groups, over the course of four chapters and a bit more than 100 pages, the book starts from scratch with the definition and proceeds through some of the more advanced topics: groups operating on sets, Sylow theory, nilpotent and solvable groups, free groups.

As for rings, there are two chapters on, respectively, the basic theory of rings and modules, followed by chapters on factorization theory (UFDs, PIDs, etc.) and modules over PIDs (including the standard applications to canonical forms in linear algebra and the classification of finite abelian groups). There is, also in common with the preceding, a chapter on noncommutative rings, culminating in the Weddeburn-Artin theorem, two forms of which (one on completely reducible rings and the other on semiprimitive Artinian rings) are given. The book ends with a chapter on tensor products that introduces some basic notions of category theory.

Between the chapters on modules over a PID and noncommutative rings, there is a chapter on fields. In the span of about 60 pages, the authors give an efficient presentation of Galois theory and also discuss transcendental extensions. This, too, is material that is generally covered, though perhaps not in the exact same way, in the books listed above.

And this brings us to some of the *differences* between these books and the one now under review. Perhaps the most striking distinction is that this text has a strong combinatorial bent, which manifests itself in several ways. First, before ever getting to group theory, the authors devote a substantial chapter to combinatorial aspects of algebra, specifically things like partially ordered sets and lattices, axioms for an abstract dependence relation, and matroids. The material here is done at a fairly sophisticated level and is referred to frequently in the rest of the book. Instructors with a combinatorial mindset (i.e., those who think of the Jordan-Hölder theorem as *really *being about semimodular lower semilattices) might find this an excellent way of doing things, but others may think that for a first look at graduate algebra, a more traditional approach is desired.

Combinatorial ideas are used elsewhere in the text as well. Graph theory, in particular, shows up in several places; the free group on a set X, for example, is defined in a fairly nonstandard way as a group of automorphisms of a particular graph; in addition, graph considerations are used to prove several algebraic results, such as a theorem of Feit, Lyndon and Scott on transpositions in the symmetric group.

There also seemed to be more of an emphasis on monoids than one might typically find. The term is defined early and used throughout the text, including, even, in the definition of a ring (although the authors do immediately then summarize the properties of a ring without using that term). In contrast to Hungerford, but in keeping with most of the more modern accounts of the subject, the authors’ definition of a ring includes the requirement of a multiplicative identity.

A number of topics that are sometimes, and in some cases often, found in competing graduate algebra texts are not covered here. The omission that I most regretted was that of group representation theory, an area that I think is becoming increasingly important and valuable for graduate students to have some knowledge of. I am apparently not alone in this belief: of the competing texts mentioned in the first paragraph, only Hungerford omits this topic. (Even some books that are, at least nominally, intended for an undergraduate audience, such as Artin’s *Algebra*, cover this topic.) Fernando Gouvêa, author of *A Guide to Groups, Rings and Fields* (an excellent overview of a graduate algebra course, but omitted from the first paragraph because it’s not really a text) writes there: “No survey of the theory of groups is complete without a look at the theory of group representations.” I agree.

There are other omissions. Basic algebraic geometry, for example, is not covered here to the extent that it is in most books at his level. Although the correspondence between polynomial ideals and algebraic varieties is mentioned, and Hilbert’s Nullstellensatz is stated, it is done so in passing and without proof (and also not by that name; it is called the “zeroes theorem”). Literally all of the books mentioned in the first paragraph above prove this result (in the case of Jacobson, it’s in Volume II). Gröbner bases are not mentioned here at all; this topic is, admittedly, less commonly covered in introductory graduate algebra texts, but it does appear in Dummit/Foote, and the inclusion of this topic there is in fact mentioned as the primary change from the second to the third edition.

Finally, instructors who really enjoy the homological aspects of algebra will no doubt also miss the inclusion of Ext and Tor from this text.

Other than content comparisons, how does the book stand up on its own? For the most part, reasonably well: the writing style is generally clear, and there are a reasonable number of examples (63, to be precise, some of which have multiple parts; the authors number them consecutively throughout the text), though on occasion I would have liked to have seen more, particularly in conjunction with a topic like Sylow theory, which practically begs for the inclusion of lots of worked-out examples.

As for exercises, there are nowhere near as many here as there are in Dummit/Foote, but there are an adequate number, some of which develop fairly well-known results (e.g., the formula for the number of irreducible polynomials of degree n over a field with q elements, and many properties of the Jacobson radical).

There are also occasional moments of carelessness, such as in Lemma 3.2.3 (subgroup criterion), which omits the requirement that the subset \(X\) be nonempty. Another example is exercise 3 on page 101, which asks the reader to show that if \(Z\) is the center of \(G\), the quotient group \(G/Z\) can never be cyclic; obviously the authors mean “nontrivial cyclic”, since for abelian groups \(G\) this quotient group obviously *is *cyclic, of order 1. Actually, this exercise points out a certain awkwardness with the way in which the authors treat cyclic groups in general. These seem to be first mentioned on page 75, where we are told that the “cyclic group of order n” can be thought of as the group of rotations of a regular n-gon (ignoring the troublesome cases \(n = 1\) and \(n = 2\)), and then a little later on page 78 the term “cyclic subgroup” of a group is defined, but I never actually saw the phrase “cyclic group” defined as a group that is equal to one of its cyclic subgroups.

Likewise, although the text never, as far as I can see, defines the term “congruence modulo \(n\)”, the phrases “quadratic residues mod 7” and “integers mod 7” appear on page 122 in connection with a discussion of the Fano plane. The book also refers to the ring \(\mathbb{Z}/\mathbb{Z}n\) of residue classes modulo \(n\), but these are defined in terms of cosets. The Euler phi-function is referenced on page 189, though not actually defined (in connection with cyclotomic polynomials) until page 391, where the term “totient function” is used. What’s even stranger is that the phrases “Euler phi-function” and “Euler totient function” are mentioned separately and with different page references in the Index, despite the fact that these phrases describe the same thing. I wondered whether this was the result of having two authors of the text.

Speaking of the Index, although it exceeds eight pages in length, it should probably be at least half again as big; I often found it to be deficient. There is, for example, no reference to either “zeroes theorem” or Nullstensatz; the Hilbert basis theorem; the Weddeburn-Artin theorem; the Jordan-Hölder theorem; or any of a number of other things for which I looked (for example, I found “ideal, prime” but not “ideal, maximal”, which is defined one page previously; in a similar example, the prime radical appears in the Index, but the Jacobson radical, defined one page later, does not). Considering, for example, that the Hilbert basis theorem appears in the chapter on modules rather than in any of the ring-theory chapters, locating things in this book can occasionally be a chore. This problem, especially in conjunction with some of the omissions mentioned earlier, lends a certain irony to the subtitle of this book, which describes itself as a “source book”.

To conclude: this is an interesting take on graduate algebra, and anybody who teaches a course on that subject should certainly take a look at this book. My favorites, however, remain Dummit/Foote and Isaacs, the former for its extensive coverage and the latter for its elegance and excellent writing.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.