Larry Grove’s Algebra is a Dover re-issue of a 1983 original, so the first two lines of the book’s Preface are a bit dated: “It is fairly standard for first-year graduate students in mathematics in the United States to take a course in abstract algebra. Most, but not all, of them have previously taken an undergraduate algebra course, but the content and substance of that course vary widely.”
Well, to be sure, even today, nearly thirty years later, algebra is a mainstay of the first year graduate curriculum; however, what passes for undergraduate algebra has evidently changed a good deal in the interim. In my own undergraduate days (in the late 1970s) we started off with a quarter of general algebra, including discussion of general material on groups, rings, integral domains, and a bit on vector spaces. We used Neil McCoy’s book for this purpose, and, to be sure, all majors took this course, from future grade school teachers to future researchers. But immediately following thereupon, the focus narrowed, falling on the graduate school bound kids, and the second quarter was devoted to group theory proper, extensively using Herstein’s wonderful but beefy Topics in Algebra; and then the third quarter was devoted to rings and fields, with Galois Theory featured very heavily, and Herstein stayed in the game accordingly.
So it was that all three Sylow theorems were hit toward the end of the second quarter, right after e.g. the theorem on finite(ly generated) abelian groups and the theorem of Cauchy; I also recall doing something with the class formula at that stage. And then the third quarter built up to the Galois correspondence theorem and its various accoutrements: all in all a pretty sporty affair.
This was really the standard path in those days, at least at research-oriented departments, and this undergraduate preparation went a long way toward getting us ready for what was in store for us in graduate school. Of course, to get ready for the quals one needed a good deal of linear algebra, too (well beyond the first course: modules over a PID were called for), a lot more field theory (along the lines of the opening salvos of the theory of algebraic number fields, for example), and some homological algebra and possibly some representation theory. Some of us picked at least a chunk of this material up in undergraduate seminars or special courses (given that ours was a department with a huge number of offerings), while others went at these things in the more orthodox context of the graduate algebra sequence: I’m sure these scenarios cover most of the spectrum of experiences of my generation of mathematicians.
When I entered graduate school in the 1980s, at a place known for its bucolic potential to make a rookie believe he’d lucked out to be going to a beach resort for his PhD and, at the same time, for its ruthlessness in destroying such a misguided rookie sometimes in just a matter of months, it eventually dawned on me that, indeed, the preparation levels of entering students were rather disparate, especially in algebra. (A propos, the local seasoned veterans would of course warn the rookies coming from a user-friendly undergraduate institution, where faculty actually showed up for their office hours and department parties included undergraduate riff-raff, to take the first year graduate sequence in algebra and memorize every word: the quals’ sword of Damocles was being sharpened all the time…) In fact, what I was witnessing was the phenomenon that led to Grove’s evident motivation for the book under review.
Already in the 1980s it was becoming clear that the heterogeneity in backgrounds in algebra among incoming graduates students should be addressed in some way: the game was changing as the population of newbies was changing, and, indeed, as the academic climate was changing, and not even Mathematics was immune.
Thus, now even more so than thirty years ago, the undergraduate algebra curriculum is all too often (let’s call a spade a spade:) compromised by the fact that the pool of students has expanded so dramatically that, for example, Herstein’s Topics in Algebra is entirely out of the question as a text. As a consequence, the average rookie graduate student starts off at a far more basic level than was the case in the days of my otherwise largely misspent youth. (And, yes, I am rocking my rocking chair with righteous indignation as I write this!)
Well, my whining aside, what all this nostalgia means is that the market has never been better for a book like Grove’s: his stated goal is to address this preparation gap, and the question is, how well does he do it?
In short: he does a good job. In terse fashion (good: a rookie needs to work hard, write in the margins, doodle on scratch paper, &c. — it’s all virtuous and character-building) Grove presents the whole panorama of what even a PDE specialist should know about algebra, so to speak. But terse is really terse in this case: Sylow appears on p. 19, and Grove is classifying (certain) finite groups by p. 40. But there are lots of examples and exercises (do them!), so the rookie who takes this business to heart will do well.
All of us know the script for “grad algebra,” of course, so my delineating what’s between the book’s covers is unnecessary. I can’t resist mentioning some favorites: Hilbert’s basis theorem: p. 73; Galois Theory (= the Correspondence Theorem): p. 90 (and it’s preceded by a very thorough preamble); Dedekind domains: p. 217; and “Representations and Characters of Finite Groups” comes in at p. 260.) I do want to mention, also, that Grove’s choices of reference books for his text are outstanding, including (in his list if five main sources) four of my all-time favorites: Emil Artin, B. L. van der Warden, Zariski-Samuel, and Curtis-Reiner. His excellent taste is transmitted to the material he goes on to cover (see, e.g., his discussion of tensor products on p. 158 ff.)
So, in closing, Grove’s Algebra is a very useful and welcome addition to the set of sources from which to learn “grad algebra” and happily (in this day and age when “Green Rudin” goes for over $80, while my copy still has its $18 stamp on the inside cover) it is priced at a mere $19.95 — what a steal!
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.