This is a text for an upper-level undergraduate course in abstract algebra. In both topic coverage and method of presentation, the book is on the more sophisticated end of the spectrum.

As to topic coverage: there is more than enough material in this text for a two-semester course. All the usual topics that one might expect to find in a two-semester course are certainly here, but there is also quite a bit more. The book covers basic set theory (functions, equivalence relations, etc.), number theory of the integers (divisibility, prime factorization, congruences), groups (quotient groups, group actions, Sylow theory, solvable and nilpotent groups) and rings and fields (rings and ideals, factorization theory, field extensions, finite fields, Galois theory). Nice applications of these ideas (e.g., compass and straightedge constructions, solution of polynomials by radicals, coding theory) are also given. There are also chapters on vector spaces (including some topics that don’t often get covered in introductory linear algebra courses, like dual spaces and triangularization of matrices), modules (including modules over a PID, with applications to the rational, Smith and Jordan forms of a matrix), and tensor products.

The chapters on modules and tensor products cover topics that I would think are rarely reached in the first two semesters of an undergraduate abstract algebra course. Even in those chapters that do correspond to topics generally associated with introductory abstract algebra courses, the material is sometimes covered in greater depth than usual. For example, chapter 3, introducing groups, also discusses the principle of inclusion-exclusion and derangements; chapter 11 on fields covers Latin squares and Steiner triple systems. Another unusual feature, at least at the undergraduate level, is a proof of the Fundamental Theorem of Algebra that uses Sylow and Galois theory and the intermediate value theorem of calculus. The necessary and sufficient condition for the constructability of a regular *n*-gon (in terms of Fermat primes) is not only stated but proved. Zorn’s Lemma is introduced at the end of the book and applications of it (the existence of an algebraic closure, the existence of a basis for an arbitrary vector space, the existence of maximal ideals in rings with identity) are discussed.

As for the method of presentation: the author’s writing style is clear and enjoyable to read, but it is also brisk and efficient. The discussion of basic number theory, for example, takes all of 14 pages, and it takes the author only about 65 pages to cover a first semester’s worth of group theory. (Actually, these 65 pages cover *more *than the standard introductory topics; they include coverage of group actions and Sylow theory as well, topics that are often deferred to a second course.) The introductory chapter on rings goes from the definition of “ring” on page 100 to a statement and proof of the Hilbert basis theorem on page 118. This is, therefore, not a book for students who need a great deal of hand-holding.

The exercises are good; around ten of them (give or take a few) appear at the end of every section (not chapter), and cover a reasonable range of difficulty. Simple verification problems (is such-and-such a group?) appear, but so do more conceptual problems asking for proofs. Based on a quick perusal, I didn’t notice any that appear inordinately difficult. No solutions are provided, but sometimes the author offers a hint.

To summarize and conclude: because of the choice of topics covered and the succinctness of exposition, this book is probably not an optimal choice as text for an “average” introductory course in abstract algebra. At the same time, because it omits a lot of graduate-level topics (group representation theory, noncommutative ring theory, categories and functors, projective and injective modules, etc.) this book likely will not replace such standard fare as Dummit and Foote’s *Abstract Algebra* as a text for a first-year graduate course. It is, however, a nicely done book, and might function well as a text for an honors course or an undergraduate course at an elite university. I would also recommend it to a student who was planning to go on to graduate school and wanted to do some independent reading to help prepare for that.

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