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Advanced Modern Algebra

Joseph J. Rotman
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 114
BLL Rating: 

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
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The new edition of Rotman's graduate algebra text comes from a different publisher at a slightly lower price. See Marion Cohen's review of the first edition, with which I concur: this is an excellent book containing much more than what is likely to be covered in a standard graduate course. It certainly fulfills the author's vision of a book that contains "many of the standard theorems and definitions that users of Algebra need to know." (The "many" in that sentence is characteristic of the author's careful writing.)

Rotman has completely rewritten the book for the new edition. Some of the changes are:

  • Rather than beginning with a review of basic algebra, the book now simply refers when necessary to Rotman's A First Course in Abstract Algebra. The "Things Past" chapter is gone, but "Groups I" and "Commutative Rings I" are still there and contain most of what is typically covered in an undergraduate course.
  • Perhaps responding to criticisms of the first edition, the author has given a bit more emphasis to noncommutative algebra: noncommutative rings are introduced earlier, division rings and the Brauer group now appear just after the Wedderburn-Artin theorem.
  • Several topics have been added, including infinite Galois theory and Abelian Categories.
  • The index (one of the few things Marion Cohen found substandard in her review) seems to have been much improved.

The best features of the first edition are retained, including Rotman's humane and elegant approach to mathematical exposition: things are explained in both words and symbols, there are historical (and even autobiographical) remarks, and the etymology of some unusual terms is explored. Most importantly, the author often takes the time to put on paper the kind of "here's how to think about it" advice that mathematicians often share with each other only orally.

In the introduction, Rotman says that "each generation should survey Algebra to make it serve the present time." His Advanced Modern Algebra admirably fulfills that goal.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and the editor of MAA Reviews.


See the table of contents posted at the AMS website.