Algebra I

The term “algebra”, as used in the title, refers to both abstract and linear algebra, both of which the text develops, at a rather rapid pace, to a fairly sophisticated level. The one-paragraph preface describes this as a text for “the first part of an intensive 2-year course of algebra for students beginning a professional study of higher mathematics.” A second volume (Algebra II), covering more advanced topics such as multilinear algebra, representation theory (not just of groups, but also Lie algebras), algebraic geometry and Galois theory, has recently been published. According to the back covers of both volumes, they are intended for “an intensive ‘Russian-style’ two-year undergraduate course”, but the “undergraduate” reference should not, I think, lead anybody to believe that either text could be used at that level at most universities in this country. This is especially true, of course, with regard to Algebra II, but it is also true of Algebra I, the book now under review.

The selection of topics covered in Algebra I is, by the standards of typical American texts, quite idiosyncratic, as is the order in which these topics are covered. This is noticeable even in the very first chapter, which, like many other first chapters in abstract algebra textbooks, covers the basics of sets, functions, and relations. Here, however, the author mentions a number of topics not typically covered in such a chapter, such as Young diagrams. In addition, even before the term “group” has been defined in general, the author defines and discusses transformation and permutation groups.

At this point, conventional wisdom would dictate that the next chapter be on elementary number theory, and that is the case here, but with a twist. Before discussing the integers, the author defines and discusses fields, rings and abelian groups, and discusses the concept of divisibility in the more general context of commutative rings with identity. Thus, much of the general theory of integer number theory is developed from this more general point of view.

Chapters 3 through 5 explore topics in ring theory. Chapter 3 discusses polynomial rings (and rings of formal power series) and uses them to construct fields by mimicking the concept of congruence for polynomials and defining residue class rings of polynomial rings modulo an irreducible polynomial. The notation used here anticipates the concept of quotient rings that will be shortly be defined. Chapter 4 is an unusual one, covering some topics not often found in books at this level: beginning with a discussion of localization in general commutative rings, it moves on to such novel topics as fractional power series and Puiseux series. Chapter 5 is a succinct but reasonably standard account of ideals and quotient rings, Noetherian rings and factorization theory.

There follows a series of five chapters that address linear algebra at a rather high level. Specifically, the topics covered include vectors, duality, matrices, determinants and Euclidean spaces. However, here again, the author takes the time to introduce a number of topics that are not generally found in more standard treatments of these subjects. The chapter on vectors, for example, contains a substantial section on affine spaces (unfortunately, as in Linear Algebra and Geometry by Shafarevich and Remizov, the treatment is heavily algebraic and actual geometry is downplayed).

Chapter 11 is titled “Projective Spaces”, and while it does contain material on those objects, including a discussion of cross ratio, it also constitutes an introduction to the algebraic geometry of affine and projective varieties. Results like the Nullstellensatz are deferred to the second volume, however.

There follow two chapters on group theory. Although the author has previously spoken of abelian groups and transformation groups, the general notion of a group is defined for the first time in chapter 12. Perhaps it is just a question of an old dog finding it hard to learn new tricks, but I thought this arrangement of material was a bit odd. The exposition here is again succinct and rapid: within the space of about 20 pages, for example, we have gone from the definition to a discussion of group actions, orbits and stabilizers. And speaking of odd arrangement of topics, it should be noted that the discussion of orbits and stabilizers, including a proof of Burnside’s formula and applications to combinatorics, comes before what I, at least, would consider to be be considerably more elementary topics (e.g., cosets, normal subgroups and quotient groups). The next chapter explores group theory in more depth: e.g., generators and relations, Sylow theory, Jordan-Hölder, semidirect products.

Chapter 14 is on the theory of modules over a PID, with applications at the end to finitely generated abelian groups. The other standard application of this theory, of course, is to canonical forms of linear transformations, and that leads to chapter 15 on linear operators, the first of five chapters discussing topics in linear algebra that are rarely taught to undergraduates. Chapter 15 addresses eigentheory, diagonalizability, the minimal and characteristic polynomials, and the Jordan canonical form. (The rational canonical form is not discussed.) Some indication of how these topics may be approached analytically is also given. Chapters 16 and 17 discuss bilinear forms and quadratic forms. The discussion here is also at quite a high level, and covers topics, such as the Pfaffian and Witt cancellation theorem, that are often found only in more specialized books on the subject, such as Lam’s Introduction to Quadratic Forms Over Fields. Chapters 18 and 19 address complex vector spaces, and here again the treatment is quite eclectic: the author discusses, for example, the difference between real and complex differentiability and deduces the Cauchy-Riemann equations of complex analysis by a matrix argument.

In a final chapter, quaternions are introduced. (It is not until this chapter that the notion of a division ring is defined.) The geometric and topological aspects of quaternions, including their relationship to matrix groups and spinors, is explored in somewhat more detail than is usual for introductory algebra textbooks.

I have discussed the contents of this book in somewhat more detail than may be customary, primarily to illustrate a comment made earlier: there is a lot of mathematics involved here, much of it beyond the scope of anything resembling an undergraduate course at an average university. At the same time, there are graduate-level topics that an instructor might want to cover (for example, group representation theory, projective and injective modules, and topics in commutative algebra such as the Nullstellensatz) that are not covered here, so this book’s use as a graduate level text is also problematic.

All in all, I think that this is a book that is more likely to appeal to professionals than students beginning their study of algebra. I think the book should make a nice reference for faculty members, but, based on the topics covered, the order in which they are covered, and the speed at which they are covered, I can’t see using this book in a classroom.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.