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Algebra II

Alexey L. Gorodentsev
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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This is the second of a two-volume set of books offering a “Russian-style” intensive introduction to abstract algebra. For a discussion of the contents of volume I, see our earlier review of that book. As with the first volume, the back-cover blurb of this text advertises it as undergraduate level. This is a statement that, for the first volume, had to be taken with a large grain of salt, and for this one, I think, should be disregarded altogether, perhaps with a snort of derisive laughter. In both volumes, the choice of topics covered and the way in which the topics are covered are decidedly unsuitable for undergraduate students at any average American university. In fact, trying to learn algebra from this text is like, as the old saying goes, trying to drink from a fire hose.

The back cover of this volume (as well as the first) also contains the following sentence: “The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry — topics that are often overlooked in standard undergraduate courses.” What the book doesn’t say is that there is a reason why these topics are often overlooked — they are, for the most part, difficult subjects that are beyond the ken of most undergraduate mathematics majors. Some of these topics (notably Galois theory) can be taught to undergraduates successfully, but doing so generally involves a certain amount of textual hand-holding and lots of examples, and that is not the modus operandi of this book.

The practical reality of budget and time constraints must also be considered here. Most universities offer, at the undergraduate level, only two semesters of abstract algebra. The first is usually devoted to teaching the students the basics of groups, rings and fields, and the second generally tries to expand on that knowledge with more advanced topics, often ring theory (factorization theory, field extensions and, with any luck, a first look at Galois theory). The number of mathematics majors at an average university that have the time and talent to tackle more sophisticated topics in algebra is probably small enough that a course based on those topics would likely not have sufficient enrollment to run.

It is precisely these “more sophisticated topics” that occupy the bulk of the book under review. The first two chapters are on tensor products of modules and tensor algebras. This is followed by a chapter on symmetric functions and another chapter on the “calculus of arrays, tableaux and diagrams”; the latter is very combinatorial and discusses some material that I had seen before, such as Young diagrams, but also quite a lot of material, such as DU-sets and combinatorial Schur polynomials, that was completely new to me. Needless to say, this is not material that is typically found in other algebra textbooks, either at the undergraduate or graduate level.

Chapter 5 is the first of four chapters discussing representation theory — not just groups, but also Lie algebras (specifically, \(sl(2)\)-modules). Semisimple modules also appear here.

Chapter 9 is on categories and functors, with quick looks also at presheaves, simplicial complexes, limits and colimits, and other assorted notions of homological algebra. (Some understanding of what a topological space is appears to be assumed on the part of the reader.)

The next three chapters deal with aspects of commutative algebra. Chapter 10 discusses integral extensions of a commutative ring with identity and algebraic and transcendental elements. The notion of a transcendence base is introduced, and Luroth’s theorem is stated and proved, after which there are two chapters introducing algebraic geometry, including sheaves. The book then ends with two chapters on algebraic and finite field extensions and Galois theory.

Each chapter contains both “exercises” that are imbedded in the text, and “problems” that are collected at the end of each chapter. The latter tend to be more substantial, and often develop topics and well-known results that are not covered in the book. For example, one problem essentially leads the student through a proof of the Weddeburn-Artin theorem. Some problems are simply unreasonable; one, for example, calls for a compass-straightedge construction of a regular 17-gon. Hints for some, but not all, of the problems appear at the end of the book.

The exposition in this text is terse and rapid, and examples are, to put it mildly, not thick on the ground. In chapter 8 on \(sl(2)\)-modules, for example, the definition of a Lie algebra is followed by exactly one example (admittedly, a rather general one): the Lie algebra formed by the commutator operation on an associative algebra. The simple example of an abelian Lie algebra (define the bracket operation to be identically zero) is not even mentioned explicitly. And then, having given the one example that he does, the author immediately plunges into a discussion of universal enveloping algebras — a topic that is sufficiently advanced that it doesn’t even appear until about halfway through Humphreys’ classic text.

Also, if any topic in algebra practically begs for examples, it is Galois theory, but again there is a dearth of them here. Although chapter 14 is entitled “Examples of Galois Groups”, it is really a theoretical chapter that discusses applications of Galois theory to such topics as compass and straightedge constructions, cyclotomic extensions and solvability by radicals. What I would have liked to see, and did not, is, right after the definition of a Galois group or the discussion of the Fundamental Theorem of Galois Theory (which, by the way, although stated and proved, is not given that name) a detailed example of the calculation of a Galois group and an explicit verification of the correspondence between the subgroups of that group and the subfields of the relevant extension field. It would simply never occur to me, for example, to discuss the Fundamental Theorem without spending a half-hour or so going through an example like the splitting field of \(x^3-2\) in detail.

As with the first volume, the order of presentation of the material often struck me as odd. For example, field extensions and Galois theory, topics that are certainly important in any first year graduate algebra course and, as noted before, are occasionally actually taught to undergraduates, appear at the very end of the book, after a great deal of much more complicated, graduate-level, material (described above) has already been done.

Even within a particular topic, the arrangement of material does not seem pedagogically optimal. Take, for example, the chapters on representation theory. If I were teaching representation theory to undergraduates, I would begin with something concrete, say specific examples of representations of finite groups, and then generalize. In this text, however, the discussion of representation theory begins with topics like tensor products and semisimplicity, and only after a chapter of that does the author get to representations of finite groups. As a result, for example, the notion of a group character comes after a discussion of Pontryagin duality.

For all these reasons, this book is not suitable as an undergraduate text for a course in an average university in this country. The question then arises as to whether it could function as a text for a graduate course. For a typical one-year course of this nature, starting from scratch, both volumes would be required, as neither one by itself contains all the material that would generally be covered in such a course. And since there are many other very good books that could function as a stand-alone text (Dummit and Foote being the obvious example), there seems to be no particular reason to use two books when one will serve, particularly if that one is better suited for student readers.

In summary, this book, like its predecessor, seems to have been written for mathematicians rather than students; I do not believe that a typical undergraduate could derive much benefit from this book, but faculty members may find it a useful reference.

Mark Hunacek ( teaches mathematics at Iowa State University. 

See the table of contents in the publisher's webpage.