For several years now I have had a friendly disagreement with a colleague of mine here at Iowa State University over the best way to teach our introductory abstract algebra course. He likes to spend a lot of time during the first semester teaching applications of the subject — cryptography, coding theory, the Fifteen puzzle, etc. He believes that the students’ fondness for the computational, concrete aspects of these applications increases their enthusiasm for, and interest in, the material.

My feeling, by contrast, is that applications are best presented after the student has thoroughly assimilated the basic ideas; at this level I am more interested in having the students develop some intuition for the basic constructs of abstract algebra and also develop proof-writing skills. Too often, I find, students leave a first-semester course without really understanding what, for example, a quotient group is, and I don’t see that looking at computational applications helps cement that kind of “gut understanding” in the way that looking at lots of examples of quotient groups does. In addition, I have noticed that I rarely have time to cover all the topics I would like to, and spending time on applications just increases that problem; my friend, for example, never even gets to the definition of a ring when he teaches the course, an outcome that I view as unacceptable.

Disagreements like ours are, of course, the reason why undergraduate abstract algebra textbooks come in a variety of flavors. Just about all of them have at least a couple of easily-skipped sections on applications so as to provide some flexibility for the instructor, but some make applications the centerpiece of the book. As the title of the book under review indicates, this book is of the latter type. The applications here lean toward the cryptographical and coding-theory variety; in that sense, the applications are not just of algebra but also number theory as well. Indeed, there is an entire chapter on cryptography before we even get to the chapter on groups.

In more detail: the first two chapters discuss the rudiments of elementary number theory (divisibility, primes, congruence modulo \(n\) and the field \(\mathbb{Z}_p\), the theorems of Euler and Fermat, and the expression of integers in bases other than 10) and its applications to cryptography (the classical cyphers, trapdoor functions, RSA, and a look at complexity of algorithms). The terms “commutative group” and “ring” are briefly defined, but the abstract theory of groups and rings is not developed at this point.

The next chapter is on group theory. In a nod to history, groups are looked at first as groups of permutations (with an application to the Fifteen Puzzle), and then the abstract definition is given. (Surprisingly, the author does not refer back to the definition of “commutative group” that was given earlier in chapter 1.) From here, the author proceeds quite efficiently from the very basic introductory stuff (subgroups, isomorphisms, etc.) to a discussion of the group of points on an elliptic curve (not surprisingly, a number of details are stated without proof).

Next come two chapters that cover, respectively, fields and polynomials. In the chapter on fields, the author proves that any finite field has prime-power order and also proves that the multiplicative group of a finite field is cyclic. (Background material on linear algebra is developed to the extent necessary, but this material can be easily omitted by students who are already familiar with the rudiments of vector spaces.) It is also stated, but not proved, that any two finite fields of the same order are isomorphic.

The chapter on polynomials begins with a less-than-precise definition of a polynomial as a formal “expression” (I am not a fan of this approach; see my review of Miller’s *Essentials of Modern Algebra*) and uses polynomial rings to produce finite fields. Although this chapter introduces polynomials, they were actually mentioned earlier in the text; the preceding chapter on fields, for example, pointed out that every element of the finite field with \(p^n\) elements is a root of the polynomial \(x^{p^n}-x\).

By this time, sufficient algebraic machinery has been developed to discuss more applications, and the author finishes off the main body of the text with three chapters on secret sharing, error-correcting codes, and compression. The first and last of these chapters are fairly short (about 15 pages long); the chapter on codes, at about 40 pages, is substantially longer. The topic of compression, a topic that seems to be getting more and more popular, is a welcome addition to this text; the older books on my shelf that discuss applied abstract algebra, such as *Applied Abstract Algebra* by Lidl and Pilz and *Topics in Applied Abstract Algebra* by Nagpaul and Jain, do not cover this.

The computer algebra system GAP (Groups, Algorithms and Programming) is used throughout the text in examples, and a number of exercises also call for its use. An Appendix at the end of the book provides an introduction to this system.

Because of the heavy emphasis on applications, many topics that would normally be covered in an introductory abstract algebra course are not covered here. The word “homomorphism”, for example, does not appear in this book; neither does the phrase “normal subgroup”. Likewise, the general definitions of “ideal” and “quotient ring” are not given, though the quotient ring of the ring of polynomials \(F[x]\) by the set of multiples of an polynomial is shown (without using the term “quotient ring”) to be a commutative ring with identity, and a field if and only if the polynomial is irreducible. This construction is done, however, with no real indication that it is a special case of something more general. The general tenor of this book, in fact, is to look at concrete algebraic objects (e.g., polynomials and elliptic curves) rather than develop general algebraic theory.

Because of these omissions, this text is not suitable for a general introductory course. However, an instructor who is willing to cover the bare minimum of algebra as a trade-off for covering applications in detail, will certainly want to look at this book. The author’s writing style is generally quite clear, and there are an adequate number of exercises and examples. Solutions to all the exercises do appear in a lengthy (about 75 page long) section at the end of the book, which may make life a little more complicated for an instructor using this book as a text (but which also enhances the value of this book for a person using it for self-study).

And speaking of making life complicated, I cannot end this review without commenting on one seriously bad feature of the text: there is *no Index at all*. I view this as an inexcusable omission. It is also, unfortunately, not the first time I have recently encountered a Springer mathematics text without an Index; see, e.g., my review of Wallis’ *The Mathematics of Elections and Voting*. If Springer thinks that omitting an Index is an acceptable (cost-saving, perhaps?) practice, it needs to reconsider that decision.

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