Sometimes, what is “basic” is in the eye of the beholder. Based on the use of this term in the title of this book and the statement on the back-cover blurb that the book was intended for an undergraduate audience, I had assumed that this text would constitute a fairly traditional introductory account of abstract algebra. In fact, however, although this book is “basic” in the sense that it starts from scratch and does not assume any prior background in abstract algebra, it covers *far *more material than is generally taught in an American undergraduate abstract algebra course at any but the most elite universities. In fact, this fairly thick (more than 600 pages long) book covers a lot of topics that are not generally taught to American undergraduates at all.

We find here, for example, sections on graduate-level topics like topological groups and Lie groups, homology and cohomology modules, the topology on the prime spectrum of a ring and other aspects of algebraic geometry, and (in an appendix) the rudiments of category theory. There is also a lot of material that, even though it might be arguably be presented to students at some point in their undergraduate education, usually comes quite a bit after a basic course in abstract algebra. Examples of this include a fairly extensive look at elementary number theory, including, for example, a discussion of complexity of algorithms and primality testing.

Here is a more systematic summary of the book’s contents. The book opens, as do many texts in abstract algebra, with a chapter on sets and functions, but this chapter takes the material considerably further than is common in books at this level, going into, for example, basic cardinal arithmetic (including a statement and proof of the Schroeder-Bernstein theorem) and even a brief discussion of the continuum hypothesis. Little subtle points that are often overlooked in undergraduate textbooks (such as the fact that any two null sets are equal) are mentioned here as well. This chapter also contains, as is again common, a section on the integers, but here again the material is presented in more depth than is customary: Peano’s axioms are discussed, for example, and the operation of addition is defined in terms of a successor function.

Group theory is the primary subject of the next two chapters, which take us from the definition of “group” through group actions and the Sylow theorems. As with the first chapter, though, there is quite a lot of material thrown in here that is very nonstandard for an undergraduate course in algebra (or any undergraduate course, for that matter). Before getting to the definition of a group, the authors discuss groupoids, semigroups and monoids, for example. This is unusual, but not unheard of; Jonathan Smith’s *Introduction to Abstract Algebra*, for example, does semigroups and monoids before groups. What *is* almost unheard of, though, is the extent to which topology is discussed in these chapters: topological semigroups are discussed, as is the fundamental group of a topological space; Lie groups and topological groups are also discussed. Applications of these ideas to computer science, by way of semigroups acting on state machines, also appear.

Some prior background in topology seems to be assumed by the authors, but the assumption is unevenly applied. The term “metric space” is used on page 69, for example, but not formally defined until page 330. Pages 68 and 69 also contain words or phrases like “Hausdorff,” “topology,” and “locally compact” that appear not to have been previously defined in the text. It would seem, however, that these early topological discussions can be safely omitted without serious consequence by anybody who is not already familiar with the ideas involved.

The next four chapters address various aspects of ring theory, and again, although the book starts at the beginning with the definition of “ring,” it proceeds at a brisk clip through some fairly sophisticated uses of ring theory, primarily to commutative algebra and algebraic geometry. Specifically, after discussing the basics of rings and ideals, the text proceeds to discuss Noetherian and Artinian commutative rings, factorization theory, maximal and prime ideals (including Cohen’s theorem and an application thereof) and the rudiments of algebraic geometry. Topology is not ignored here either: the spec topology is discussed, and in another section the maximal ideals of the topological space C(X) (continuous functions on a compact topological space X) are studied.

Chapters 8 and 9 are rather full discussions of vector spaces and modules, respectively. The vector space chapter really amounts to a rapid survey of an entire undergraduate linear algebra course and perhaps even a little bit more: it covers vector spaces, linear independence and dependence, bases, matrices, linear transformations and their relationship to matrices, inner product spaces (including a brief section on Banach and Hilbert spaces), the Jordan canonical form, and quadratic forms (with Sylvester’s Law of Inertia). The chapter on modules includes not only the standard introductory material but also the theory of finitely-generated modules over a PID, as well as an introduction to some ideas of homological algebra (e.g., exact sequences and homology modules). Chain conditions for modules are also discussed.

The remaining three chapters of the book (10–12) continue the material on number theory that was initiated in chapter 1. They essentially comprise what seemed to me to be a full semester’s worth of material on number theory, done from an algebraic standpoint. The first of these chapters covers divisibility and prime factorization, multiplicative functions, congruences, quadratic residues (including a proof of the Law of Quadratic Reciprocity), and primality testing (including a more detailed than usual discussion of complexity of algorithms). As noted, algebra is used throughout to give full insight to some of these ideas; to give just one example, Fermat’s Little Theorem is seen (as it should be) as a consequence of Lagrange’s Theorem on finite groups.

The algebraic approach is applied even more extensively in the next chapter, which provides an introductory look at field extensions and algebraic numbers. Here again, there is some unusual material presented here, including a topological proof (using homotopy) of the Fundamental Theorem of Algebra. Finally, chapter 12 gives, as an application of the number theory and algebra that has been developed, a fairly detailed look at cryptography, starting with the classical cryptosystems (affine ciphers, the Hill method, etc.) and proceeding through the more modern “public key” methods such as RSA and ElGamal. This chapter concludes with a fifteen-page introduction to the freely available software SAGE.

The book concludes with three appendices. Besides the one on category theory referred to earlier, there is one on the structure of semirings and another, entitled “A Brief Historical Note”, which starts with a very rapid overview of the history of algebra and then provides a series of paragraph-long biographical sketches, in chronological order from Euler to Zariski, of people who have influenced algebra.

The exposition throughout is reasonably clear but quite succinct. On occasion it can be *too* succinct — in chapter 6, for example, the authors offer a proof of the existence of a finite field with *q* = *p*^{n} elements for any prime *p* and positive integer *n*, but the proof seems to take for granted the existence of an irreducible polynomial of degree *n* over the field **Z**_{p}. Most of the books that I am aware of prove the existence of such a field by proving that the splitting field of the polynomial *x*^{q} – *x* is the desired field, and then use the existence of that field to prove that irreducible polynomials of every degree exist; it is also possible to prove directly that an irreducible polynomial of the right degree exists, but the proofs that I have seen of this are not obvious and use ideas like Möbius inversion.

Because of the fairly rapid pace through material that is not typically associated with an introductory undergraduate course in abstract algebra, this book will probably not find much use as a text for such courses. Unfortunately, because of some *omissions* in coverage, this book probably isn’t suitable as a graduate text for algebra courses either. Galois theory, for example, is not discussed, and other topics that one might want to cover in a year-long graduate course are also not found here: group representation theory, for example, is not mentioned, and although the Weddeburn-Artin theorem on semisimple rings is mentioned in an exercise, it is not proved in the text.

Despite my reservations about the use of this book as a text, I do believe that it should be looked at by people who teach courses in algebra and number theory. This book is certainly an unusual one, filled with interesting material, and there are quite a few exercises spanning a wide range of difficulty. (A number of the exercises develop standard material: as previously noted, Weddeburn-Artin is an exercise, as is the impossibility of trisecting an arbitrary angle, or doubling the cube, with straightedge and compass.) It should therefore have considerable value as a reference for undergraduate or graduate professors, or advanced students who want to see some applications of the material they have been taught.

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