Anthony W. Knapp’s Basic Algebra aims to teach the student algebra in a somewhat different way than is the norm. Together with its sequel, Advanced Algebra, the book proposes to take an aspiring mathematician from his first exposure to algebra proper, right after calculus, through his graduate training in algebra — and a bit further, to “what [every] young mathematician needs to know” (p. xiii). Knapp says that the litmus test for topics included is whether a plenary lecturer at a mathematics meeting would presuppose the according knowledge on the part of his audience. Therefore, Basic Algebra is something like an algebra vade mecum as well as a text for a more holistic approach to undergraduate algebra. Actually Basic Algebra does a good deal of graduate level algebra, too, at least as far as what one would need for a typical PhD qualifying examination, so Knapp’s vision certainly extends beyond the first four years of university education in mathematics. (Advanced Algebra addresses more esoteric or eclectic material, starting, as it does, with “Transition to Modern Number Theory,” and ending with “Methods of Algebraic Geometry.” See p. x of Basic Algebra.)
To use Basic Algebra in a standard undergraduate curriculum would entail spreading the material over several semesters, since the book can be used for linear algebra, group theory (Chapter IV is a gem!), rings and fields, Galois theory, a second course in group theory, and even a course on modules. Presumably a more holistic pedagogical approach, in line with what Knapp seems to have in mind, would engender the “simple” manouevre of defining an autonomous two-year course titled “Algebra,” and going through the book in linear order. Of course, in the surreal world of the contemporary academy anything this simple is far too suspect to the martinets in charge to have a reasonable chance of succeeding, but hope springs eternal.
Some remarks about Knapp’s exercises are in order. In a word, they are superb. The problems to Chapter VII, “Advanced Group Theory,” for example, feature involving one of Burnside’s theorems to show that the smallest non-abelian simple group has order 60 (it is A5, of course), an n-handle-body, the Baer product, Fourier analysis (including Poisson), FFT’s, and even some representation theory. Such wonderful depth and variety is typical of all the exercise sections, and there are over ninety pages of hints toward the problems’ solution at the end of the book. À propos, the exercises on p. 298 ff. present a good, if succinct, treatment of some Lie theory, underscoring Knapp’s generalist pedagogical objectives as delineated above.
The book also has a very evocative cover, containing the diagram (cf. p. 500) attending Gauss’ construction of the regular 17-gon.
Basic Algebra is a very interesting and well-written book, and is indeed well suited for the approach to algebra the author intends, and, for that matter, usable for commonplace approaches as well.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.
Contents.- Preface.- Guide for the Reader.- Preliminaries about the Integers, Polynomials, and Matrices.- Vector Spaces over Q, R, and C.- Inner-Product Spaces.- Groups and Group Actions.- Theory of a Single Linear Transformation.- Multilinear Algebra.- Advanced Group Theory.- Commutative Rings and Their Modules.- Fields and Galois Theory.- Modules over Noncommutative Rings.- Appendix.- Index.