Jacobson’s Basic Algebra was originally published in 1974. I think my first undergraduate exposure to algebra must have happened in 1977, and it was probably in 1978 that I bought my copy. I loved it. I read through most of the first volume, helped, I am sure, by the fact that I had already had a first exposure to most of the material. I enjoyed, in particular, the incredibly efficient proofs and exposition.
Brevity is the hallmark of Jacobson’s approach. Group theory through Sylow’s theorems is covered in about 50 pages. Basic ring theory takes slightly longer. By chapter 3 (page 157) we are launched into the theory of modules, the goal being to obtain the structure theorem for finitely-generated modules over a principal ideal domain. This is done by generalizing “row-reduction” to the context of PIDs. I had already studied the more abstract proof, and was bowled over by how simple and straightforward this account is. It didn’t occur to me then that this point of view has the extra advantage of being explicit and constructive; those were questions I didn’t yet know how to ask.
The first four chapters, which make up about 60% of the book, cover most of what I would describe as “basic” algebra: groups, rings, modules, fields, Galois theory. Then come the theory of real polynomials, the classical groups, algebras over a field, and lattices, none of which can really be considered standard topics, even for a graduate course in algebra. I find the inclusion of the classical groups and a little bit of Lie theory a Very Good Thing. On the other hand, the category theory point of view, while implicit here and there, does not really feature in the book, and the non-commutative structures that have become so important recently are dealt with only lightly. There is very little that points towards number theory and algebraic geometry.
The second edition came out in 1985, and I replaced my much-used copy of BAI with a copy of the new edition. (I should have held on to both, if only to re-read the account of the theory of finite fields which Jacobson describes, in the preface to the second edition, as “a tour de force of brevity.” That section was much expanded in the second edition, with proper signposting of the important results.) It is this second edition that Dover has now returned to us.
I once tried using BAI as the textbook for an undergraduate course, and the results were catastrophic: the brevity that I so enjoyed when I read the book proved to be a barrier my students simply could not surpass. It may be “basic algebra”, but it is not an “abstract algebra” textbook. Still, it’s a great read, and I’m delighted that it’s now possible to put it in the hands of those students who can profit from it.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.