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Basic Algebra I

Nathan Jacobson
Dover Publications
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

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.



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akirak's picture

Dover has done the mathematical world a boon of historical magnitude saving these classics from opulent obscurity. At one point,the original hardcover second volume was selling for nearly 500 dollars. It is one of the "bibles" for graduate algebra courses at the better schools. One of Jacobson's students once told me he confided later in his career that even the best undergraduates at Yale struggled with learning algebra from it, feeling that the author was "unfairly optimistic" in the goals set for them. His graduate students,however,swore by it and would never consider using any other text.

In the two decades since it came out,a legion of graduate algebra texts have emerged, but I doubt any will be able to match the mastery this text gives, particularly in ring theory. As far as I know it is the only general algebra text, apart from Louis Rowen's recent two volume epic, that discusses nonassociative rings and algebras in any depth. Yes, it could use more examples. But the examples Jacobson chooses are perfect. In many ways, Rowen's text is the 3rd edition of this one. It is a text no mathematical library is complete without, and now that it's in Dover,there's no excuse not to have it.