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Advanced Modern Algebra:Third Edition, Part 1

Joseph J. Rotman
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 165
[Reviewed by
Fernando Q. Gouvêa
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The third edition of this book is very different from the previous ones. As the reviews of the first and second editions indicate, Rotman is a very good writer. Those editions were masterful reference works presenting most of graduate-level algebra. Indeed, the second edition served as one of my go-to books as I was writing my Guide to Groups, Rings, and Fields. I often found that Rotman presented the material better than his competitors.

Clearly, however, the author has felt some restlessness about his book. Between the first and second editions, there was a change in publisher and also various additions, but there was also a significant “I changed my mind” move: instead of including a review of basic abstract algebra, Rotman decided to point to his undergraduate textbook. The additions made the second edition quite big (“elephantine”, he says in the preface to the third edition), so it is not surprising that the third edition comes in two volumes. More surprising, however, is the radical reorganization.

Where the second edition is clearly a reference book, an encyclopedic account of modern algebra to which one might turn to recall a theorem or learn a particular topic, the third edition is clearly intended as a textbook. It is divided into two parts labeled “Course I” and “Course II”, which apparently correspond to the first and second graduate courses in algebra at the University of Illinois at Urbana-Champaign.

Course I has Galois Theory as its telos: it opens with the classical formulas for solving cubics and quartics, then goes on to cover some number theory and commutative algebra, the basics of group theory, and finally Galois Theory itself. It is certainly a good choice for a first graduate course, since most undergraduates are taught some algebra but many do not see Galois Theory, whose theorems are seductive enough to motivate what is essentially a review, at greater depth, of undergraduate algebra.

Course II is a course on ring theory, both commutative and non-commutative, but mostly commutative. “Having lured students into beautiful algebra,” says Rotman, they will now want to learn more. This section includes both linear and multilinear algebra, some category theory, and some commutative algebra.

Part 2 of the book has not yet appeared, but the author describes it thus:

These two courses serve as joint prerequisites for the forthcoming Part 2, which will present more advanced topics in ring theory, group theory, algebraic number theory, homological algebra, representation theory, and algebraic geometry.

This leads me to think that it is likely to be more in the style of the second edition.

Rotman is a wonderful expositor, and the two courses in this book strike me as well thought out and well presented. (There’s too much material, as usual, so that instructors can be selective.) But I miss the compendious approach of the second edition, in which topics could be developed according to their own internal logic rather than the logic imposed by pedagogy. I’ll certainly use and learn from this edition, but I think I’m going to hold on to my copy of the second edition as well.


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Fernando Gouvêa is Carter Professor of Mathematics at Colby College. He hasn’t taught graduate algebra in more than 25 years, but he’s still a fan.