Fundamentals of Modern Algebra: A Global Perspective

This is a text in abstract algebra at the beginning graduate level. It is noticeably different from most of the other standard graduate algebra texts in its size; excluding exercises, this book is roughly 200 pages long, which makes it about one-fourth the size of Dummit and Foote’s Abstract Algebra and one-fifth the size of the second edition of Rotman’s Advanced Modern Algebra. (For its third edition, Rotman’s text has been split into two parts, the first of which, according to the AMS website, is more than 700 pages long.)

Succinctness in mathematical exposition can obviously be a good thing, but I don’t think it is here. As explained below, I think this book has excised not only fat but a lot of healthy tissue as well. The topic coverage seems inadequate for its intended purpose, and, for the topics that are covered, much more could, and should, be said.

These problems are already apparent in the first chapter, on group theory. This is the only chapter in the text devoted to group theory, but it is, again excluding an end-of-chapter exercise set, only about 35 pages long, and covers only basic topics that are routinely taught at the undergraduate level: groups, homomorphisms and isomorphisms, normal subgroups.

The only moderately sophisticated topics covered in this chapter are group actions and Sylow theory, and even here the second and third Sylow theorems are stated without proof. Many group-theory topics that one might want to cover in a graduate course are nowhere to be found; for example, there is no discussion of nilpotent and solvable groups or semi-direct products, and the Fundamental Theorem of Finite Abelian groups is not proved, but is referred to in passing as a special case of the Fundamental Theorem of Finitely Generated Groups, which is also stated without proof.

The phrases “simple group” and “alternating group” are mentioned only in the exercises, not in the body of the text, and so of course there is no statement or proof in the text of the theorem that the alternating group \(A_n\) is simple for \(n \geq 5\). The normalizer of a subgroup, and the commutator subgroup of a group, are also defined in the exercises rather than the body of the text, and the class equation of a finite group is not mentioned at all. There is also no discussion of the mammoth research endeavor that led to the classification of all finite groups.

The discussion of the topics that do appear leaves much to be desired. The exposition seems hurried and unmotivated, with minimal attention paid to examples. Indeed, this is quite likely the only textbook on abstract algebra that I have ever seen that discusses Sylow theory without giving one single example in the text of a Sylow \(p\)-subgroup. There aren’t even any explicit examples given in the exercises, although in one exercise the reader is asked to find Sylow subgroups of \(D_3\).

Unfortunately, these problems are not limited to the first chapter. There are four chapters that follow (ring theory, modules, extension fields and finite fields) and although some interesting topics are covered in them, a great deal of material that is often covered in a graduate course is omitted. By way of explanation, let’s begin with a quick summary of the topic coverage of the rest of the book.

Chapter 2 on rings is about 50 pages long. It starts from scratch with the definition of a ring, and proceeds through the standard topics (ideals, prime and maximal ideals, quotient rings, etc.) to more sophisticated fare such as valuations and \(p\)-adic numbers. The next chapter, on modules, begins with a discussion of vector spaces over an arbitrary field (including the usual Zorn’s Lemma proof of the existence of a basis); modules are defined next, including free and projective (but not injective) ones, and the tensor product.

The next two chapters are on aspects of field theory. The first, on simple algebraic extensions, constitutes an introduction to Galois theory (but only in the characteristic 0 case, thus avoiding the need to talk about separable extensions) and also an introduction to algebraic number theory, including Dedekind domains, fractional ideals and the class group. The second (and the last chapter of the text) talks about finite fields: after discussing the construction of an extension field containing a root of an irreducible polynomial, the basic facts about such fields (including the existence and uniqueness of a finite field for every prime power order) are proved. A final section applies these ideas to sequences of field elements (like the Fibonacci sequence) that are defined recursively by linear relations.

Many standard topics that one might expect to see in these chapters are omitted, however. There is no discussion, for example, of modules over a PID, a topic that I think belongs in any first-year graduate algebra course. Euclidean domains are not mentioned. Noncommutative ring theory (e.g., Weddeburn-Artin, Jacobson radical) is not discussed. And speaking of Weddeburn: his theorem on finite division rings is not here and, in fact, I don’t think that the term “division ring” is defined in the text. and I didn’t see the quaternions given as an example of a ring. Ext and Tor are also missing in action. Galois theory, as previously noted, is done only in characteristic 0, and some of the more famous applications (e.g., insolvability of the quintic) are not mentioned at all. The relationship between field extensions and the classical Greek construction problems (for example, angle trisection) is also not even hinted at.

And, as with chapter 1, some topics that are covered are not done in sufficient depth. An example is the book’s treatment of group representations. In a section of chapter 2 titled “The Group of Units of a Ring”, the notion of a group character (as a homomorphism from the group to the group of nonzero complex numbers) is defined, as is the notion of a representation of G. However, the more general notion of a character of a representation \(\rho\) (defined by the trace of the matrix \(\rho(g)\)) is never mentioned at all, and no hint is given of the usefulness of this very important topic.

As with the discussion of Sylow theory, there are fewer examples given in these chapters than I would like. Noetherian rings are defined, for example, and basic results proved about them, but no example is given of a ring that is not Noetherian. Local rings are defined, but no example is given of one (and no time at all is spent explaining why anybody would care about them).

In addition, there are occasional odd stylistic choices. For example: the author states and proves a theorem that in a ring with identity, every ideal is contained in a prime ideal. The proof that he gives, though, is the standard proof of the stronger result that any ideal is contained in a maximal ideal. Why not state the stronger result as the theorem, especially since that was what was actually proved? Another example: the term “normalizer” is used in the body of the text in the statement a theorem on page 34, but is only defined in the exercises that follow on page 37; thus a reader will have to interrupt the statement of the theorem to search for the meaning of a word used in it.

Readers of the book should also note that occasionally assumptions that apply throughout a section (e.g., that all rings are commutative with identity) are stated once at the beginning of a section and thereafter not repeated in individual theorems. So, for example, we have a theorem stating that any two bases of a free R-module of finite rank have the same number of elements, without stating explicitly in the statement of the theorem that R must be commutative (and also without any example showing how this can fail if R is not). This approach is not unheard of, but can potentially cause confusion for “drop in” readers.

It may well be that it is simply not possible to write a 200-page long text on graduate abstract algebra; there is just too much to cover. The only other book that I am aware of that tried this is Wickless’s A First Graduate Course in Abstract Algebra, but, as our review of this book makes clear, it also suffers from not covering enough graduate-level material. Gouvêa’s A Guide to Groups, Rings and Fields manages to very nicely survey a full year of graduate algebra in 300 pages, but that book is, by design, not a text; it has neither proofs nor exercises, but instead concentrates on providing a helpful overview of the subject, complete with lots of examples.

The subtitle of the book under review is “A Global Approach”. The preface never says what the author means by that phrase, but if it is intended to refer to the fact that the book offers a kind of global summary of abstract algebra, then I can’t help but think that a book like Gouvêa’s does a better job, even with proofs omitted. Gouvêa’s book offers a panoramic view of many facets of the subject and how they fit together; this book, with its deficient topic coverage, falls considerably short of that objective.

Conclusion: as a text for a graduate-level abstract algebra course, I do not believe that this book represents a pedagogical advance over such standard books as Dummit & Foote, Hungerford, or Rotman.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.