Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography

The preface to this book states that it is an “introductory text on abstract algebra … [which] grew out of courses given to advanced undergraduates and beginning graduate students in the United States and to mathematics students and teachers in Germany.” This leads me to ask: how many introductory abstract algebra courses, either graduate or undergraduate, are you aware of in which a basic result like Lagrange’s Theorem on the order of a subgroup of a finite group (proved in the text on page 129) is not proved until after proofs have been given of the existence of maximal ideals in commutative rings with identity (page 26), the fact that every PID is a UFD (page 42), the fact that R[x] is a UFD if R is (page 60), the fact that the Liouville constant c= Σ 10^-n! is transcendental (page 76), the necessary and sufficient condition (in terms of Fermat primes) for the constructability of a regular n-gon (page 88), the existence of an algebraic closure of a field (page 97), the Fundamental Theorem of Algebra (pages 105-108), and the Fundamental Theorem of Symmetric Polynomials (pages 109-111)?

Not many, I’ll wager, and therein lies what I consider to be the principal drawback to using this book as a text for any introductory course in abstract algebra: a very idiosyncratic selection and arrangement of material. After an introductory chapter 1 (“Groups, Rings and Fields”) in which many terms are defined and a number of basic results stated (some, like Lagrange’s Theorem, without proof), the text proceeds to three chapters on topics in commutative rings — maximal and prime ideals, factorization theory (primes and irreducibles, PIDs, UFDs, and Euclidean domains), and polynomial rings, in that order. (In contravention of what I believe to be fairly standard convention these days, the authors’ definition of “ring” does not require the existence of a multiplicative identity.) The next four chapters introduce field theory — field extensions, algebraic closures, splitting fields and normal extensions, and applications to ruler and compass constructions.

The next five chapters address group theory, starting literally from scratch by repeating (from chapter 1) the definition of a group, and proceeding through normal subgroups and factor groups, the symmetric and alternating groups, group actions, Sylow theory, solvable (but no mention of nilpotent) groups, and, finally, free groups and presentations. This group theory material is then applied to field theory in the next three chapters, which talk about Galois theory and its applications (including a second proof of the Fundamental Theorem of Algebra). In the next two chapters, the authors discuss modules (ultimately leading up to modules over a PID) and the application of this theory to the classification of finitely generated abelian groups (viewed, of course, as modules over the ring of integers). Unfortunately, the authors do not mention another interesting and standard example of the classification theory of finitely generated modules over a PID, namely canonical forms of matrices.

Chapter 20 of the text returns to field theory and discusses integral and transcendental extensions. Proofs are given of Noether’s normalization theorem, and the transcendence of both e and π. Unfortunately, one of my favorite applications of transcendence bases (proving that the field of complex numbers has infinitely many automorphisms) is not mentioned, either in the body of the text or as an exercise. Finally, the last two chapters of the book discuss, respectively, introductory algebraic geometry (in affine spaces, up to and including the Hilbert Basis Theorem and Nullstellensatz, and some accompanying commutative algebra) and some of the algebraic aspects of cryptography (including a brief mention of elliptic curves). Although algebraic geometry and cryptography share titular billing with Galois theory, these subjects are not developed as much in the text.

If used as a graduate text for students who already have a background in undergraduate abstract algebra, the arrangement of material may not be quite as troublesome, but there are some topics which many instructors would want to cover in a year-long graduate course that are not mentioned in this book at all. Examples of these would include vector spaces and linear transformations, nilpotent groups, projective and injective modules, noncommutative rings and Weddeburn-Artin theory, and group representation theory. Even on those topics that are included, there are strange omissions: while the authors prove, for example, that any two bases for a free module over a commutative ring with identity have the same number of elements, I could not find a counter-example for arbitrary rings. On the other hand, the inclusion of a chapter on cryptography is a bit puzzling, since most graduate algebra courses of which I am aware do not have time to cover this topic, which is usually presented, if not in a course of its own, as an application in number theory courses.

Another serious drawback to the use of this book as text is the very small number of exercises. Each chapter ends with a separate section titled “exercises”, but the number of them averages out to less than eight per chapter, and quite a few of these are rather pedestrian (of the nine exercises in chapter 3, for example, two ask for a proof that something is a ring, one asks for a proof of basic properties of the degree of the sum and product of two polynomials, another asks for a proof that the usual norm on the ring Z[√(–5)] is multiplicative, and another asks for a proof of basic properties of the norm in the ring of Gaussian integers.) Contrast this, for example, with a standard graduate algebra textbook like the one authored by Dummit and Foote, which contains exercises, sometimes in excess of forty, in every section of the book. (In the Sylow theory section, for example, there are 56 exercises.)

While I cannot, for the foregoing reasons, recommend this as a potential textbook to any person teaching a standard undergraduate or graduate algebra course, I do not want to give the impression that I think this is a bad book. As the survey of topics given above should make clear, there are lots of interesting things to be found here (not very many algebra textbooks contain two different proofs of the Fundamental Theorem of Algebra, for example), so the book can certainly be profitably used as a reference and source of interesting material. In addition, if an instructor is willing to either make do with, or supplement, the meager collection of exercises, this would be a viable text for a one-semester course in field and Galois theory in which the basic material on groups and rings is assumed.

Mark Hunacek teaches mathematics at Iowa State University. After near-simultaneous acquisitions of both a PhD and a wife, he solved the “two body problem” in his family by going to law school and then becoming an Assistant Attorney General for the state of Iowa while his wife pursued a career as a mathematics professor. He is happy to report, however, that he has now retired from the practice of law and returned to the fold of mathematics teaching (but he also teaches a course in engineering law for old time’s sake.)