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Algebra: A Graduate Course

I. Martin Isaacs
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 100
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
, on

Algebra, A Graduate Course first appeared fifteen years ago, the present volume being an AMS/GSM re-issue; its reappearance testifies to its manifest success as a graduate textbook. This is a non-trivial achievement, of course, given the stiff competition in this area. I am of an age to insist, stubbornly, that Lang’s Algebra is really the benchmark in the present context, even though I always favored Van der Waerden’s (not so) Modern Algebra myself and often used MacLane-Birkhoff (> Birkhoff-MacLane) to great advantage. And then there’s Hungerford’s popular book, and Jacobson’s Basic Algebra, and the marvelous recent contributions to the game offered by Rotman (Advanced Modern Algebra) and by Knapp (Basic Algebra and Advanced Algebra). Doubtless there are other kindred texts that haven’t yet appeared on my rather parochial radar screen, but I think the preceding books present a pretty decent sized sample.

And this begs the question, why Isaacs? Well, Isaacs himself provides an answer early on, in a charming passage in the book’s preface, in which Loomis’ introductory algebra course at Harvard (in the 1960s) is cited as the inspiration for the book under review. Isaacs, as the grateful beneficiary of Loomis’ teaching, seeks to spread the wealth, not only following Loomis’ pedagogical strategy but stressing corresponding goals and objectives — and properly so. To wit, the author:

At the end of each [of thirty] chapter[s], there is a fairly extensive list of problems [, f]ew of [them] … routine … some … quite difficult. The purpose of these problems is not just to give practice with the definitions and with understanding the theorems. My hope is that by working these problems students will get a feeling for what it is actually like to do algebra, and not just to learn it. (I [= Isaacs] should mention that when I teach my algebra course, I assign five problems per week.)

Clearly, then, and greatly to his credit, Isaacs is by no means “teaching to a test” — or even a qualifying examination, really; no, he is presenting his students with opportunities to serve apprenticeships geared toward becoming actual living and breathing algebraists, or, failing that, properly algebraically schooled mathematicians in other specialties. And, after all, learning to think and work algebraically is, in an absolute sense, a consummation devoutly to be wished.

The book’s organization is not unusual, proceeding from groups and (some) rings (i.e. non-commutative algebra) to commutative rings (in Isaacs’ hands always equipped with unity), fields and Galois theory, as well as background material for algebraic number theory, algebraic topology, algebraic geometry, etc. (i.e. commutative algebra). And then the book ends on what is to my taste an extremely high note: Chapter 30 is titled “algebraic sets and the Nulstellensatz.” To be sure, it is all there in the 516 pages of Algebra, A Graduate Course, and if a neophyte graduate student completes this apprenticeship, not only should the upcoming qualifying examination be easily within reach but he’ll be chomping at the bit to start doing real mathematical work, all in keeping with Isaacs’ objective.

A few specifics about the all-important exercises, then. The problems in group theory are very reminiscent to me, certainly in flavor, of those found in the excellent text, A Course in Group Theory, by John S. Rose, a 1978 British text advertised as something of a second course in group theory, the stress falling on group actions. As far as the book’s other sections go, the exercises, again, appear to me to be familiar in flavor, but I must factor in the condition that I’ve spent many years dealing with themes in algebra as it bears on number theory as algebraic geometry (and even algebraic topology, I guess). Thus, while this suggests that Isaacs’ chose or crafted these problems carefully, he truly doesn’t waste time on the routine stuff. Perhaps a beginner were well-advised to supplement his study of Isaacs’ book with a collection of expressly routine problems from elsewhere. Maybe Herstein’s Topics in Algebra could serve in this regard, given its stratification as far as problem sets are concerned (its original orientation to (gifted) undergraduates notwithstanding).

In any event, Isaacs’ Algebra, A Graduate Course is a pedagogically important book, to be highly recommended to fledgling algebraists — and every one else, for that matter.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.


Part One, Noncommutative Algebra 

  • Definitions and examples of groups
  • Subgroups and cosets
  • Homomorphisms
  • Group actions
  • The Sylow theorems and $p$-groups
  • Permutation groups
  • New groups from old
  • Solvable and nilpotent groups
  • Transfer
  • Operator groups and unique decompositions
  • Module theory without rings
  • Rings, ideals, and modules
  • Simple modules and primitive rings
  • Artinian rings and projective modules
  • An introduction to character theory

Part Two, Commutative Algebra 

  • Polynomial rings, PIDs, and UFDs
  • Field extensions
  • Galois theory
  • Separability and inseparability
  • Cyclotomy and geometric constructions
  • Finite fields
  • Roots, radicals, and real numbers
  • Norms, traces, and discriminants
  • Transcendental extensions
  • The Artin-Schreier theorem
  • Ideal theory
  • Noetherian rings
  • Integrality
  • Dedekind domains
  • Algebraic sets and the nullstellensatz
  • Index