This slim book (180 pages of text) reimagines

the traditional introductory course in abstract algebra. At first glance, one might think that the selection of topics is similar to that found in standard “rings first” algebra books such as Hungerford’s *Abstract Algebra: An Introduction*, but a closer look reveals substantial differences.

Hungerford’s book, for example, begins with a couple of chapters developing the basic number theory of the integers (up to and including congruences), followed by the introduction of abstract rings. This is then followed by several chapters discussing polynomials over a field as another extended example of a ring. After this, there is some more ring theory (ideals and quotient rings), followed by an introduction to groups. This constitutes Part 1 of the book. Part 2 concerns more advanced topics, much of which is not typically covered in a first semester course: the fundamental theorem of abelian groups and Sylow theory, factorization in integral domains, field extensions and an introduction to Galois theory. Part 3 of the book, on applications, considers such topics as cryptography, coding theory and geometric constructions. A fourth and final part collects a number of appendices on background topics.

The book now under review also starts with the number theory of the integers (spread out over the first three chapters), but takes the subject considerably further than does Hungerford: the law of quadratic reciprocity, for example, is discussed and proved, a rarity in introductory abstract algebra books. Other number theory topics covered here, but not in most introductory abstract algebra texts, include the classification of Pythagorean triples (done two ways, algebraically and geometrically), the proof of Fermat’s Last Theorem for n = 4, and the classification of the solutions of Pell’s Equation.

It is not just the ring of integers that is discussed in these chapters. The axioms for an abstract ring are introduced very early (page 2, in fact) and examples of rings other than the integers (for example, quadratic extensions of the integers) are introduced and worked with in some detail in these chapters. However, the focus remains on concrete examples rather than abstract ring theory per se.

Chapter 4 is on what the authors call “codes” but what I would call cryptography; I think the use of the word “codes” here risks confusion with coding theory (Hamming distance, etc.). The authors do not tarry with classical cryptography and proceed almost immediately to public key cryptography and the RSA method. Primality testing and factorization are also discussed briefly.

In Chapter 5, the authors discuss the real and complex numbers. Assuming the rational numbers as known, the authors explain without detail several methods of constructing the real numbers and also point out the completeness property of this number system. The complex numbers are then introduced and, again surprisingly for a book at this level, the Fundamental Theorem of Algebra is proved. Applications of the Fundamental Theorem to real polynomials are given.

Chapters 6 and 7 concern polynomial rings and their application to the construction and analysis of finite fields. Here again, a number of very unusual topics for a book at this level are discussed, including transcendental numbers (Liouville numbers and a proof of the transcendence of \( e \)), formulas for the solution of a cubic equation, and Sturm’s Theorem, a result I used to see in old-fashioned books on the theory of equations, but haven’t seen in the more recent textbook literature. (Another result along the same lines, Descartes’ Rule of Signs, shows up here as an exercise.)

One topic that is not covered in this book, however, is group theory. While the word “group” is mentioned and the term “abelian group” briefly defined, the subject is not developed, and the reader will not encounter even such basic ideas as Lagrange’s theorem.

Throughout the book, the exposition is rather terse, and very little hand-holding takes place. As the preceding summary of topics covered in this book should make clear, a considerable amount of material is covered in a relatively small amount of space. Each section ends with an assortment of exercises (averaging around 6 or 7 per section), few of which are trivial and some of which are quite substantial. Each chapter ends with a page or two entitled Notes which offers some historical commentary and suggestions for further reading.

I’m an old dog and therefore not easily taught new tricks, but it seems to me that the topics covered here are not optimal for a first course in abstract algebra. More adventurous souls, however, might enjoy the experience of teaching out of this book, particularly to students of high ability.

Mark Hunacek (mhunacek@iastate.edu) is a Teaching Professor Emeritus at Iowa State University.