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Advanced Modern Algebra, Third Edition, Part 2

Joseph J. Rotman
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 180
[Reviewed by
Fernando Q. Gouvêa
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See my review of Part 1, which also contains links to reviews of the previous editions of this very good book. As I noted in that review, for the third edition Rotman had decided to rearrange Part 1 of his Advanced Modern Algebra, making it more of a textbook. Hence, Part 1 is divided into two “courses,” one on Galois theory (course A) and one on ring theory (course B). Part 2 was planned to be more of a reference book, covering several algebraic subjects that didn’t fit into those two courses: “advanced topics in ring theory and group theory, algebraic number theory, homological algebra, representation theory, and algebraic geometry.” Here is that book, unfortunately published after the author’s death and not having fulfilled that promise completely.

Rotman continues the lettering from the previous Part, labeling the chapters in Part 2 C-1, C-2, etc. Chapter C-1 gives us more group theory, about 100 pages’ worth. Chapter C-2 deals with group representation theory, focusing mostly on the semisimple case. Chapter C-3 is homological algebra, with applications to what came before, and Chapter C-4 extends the basic category theory that was introduced in Part 1, focusing on additive and abelian categories, sheaves, and K-theory. Chapter C-5 seems a refugee from the second edition: it is called “Commutative Rings III” (the second edition had chapters called Commutative Rings I and II).

Alas, the chapter on algebraic geometry was never written; all that the editor gives us is a paragraph meant for the introduction to that chapter. He describes it as a “small memorial to what might have been” — it’s a beautiful discussion of “mathematical folklore” as “the mathematics everyone knows,” pointing out how this changes over time. My guess, which of course might be completely wrong, is that Rotman was planning to argue that in today’s mathematics basic algebraic geometry is part of what “everyone knows” and that therefore all graduate students should learn.

This is a good book and a fitting tribute to a great mathematical expositor. I’ll keep both the second edition and the two parts of the third edition on my shelf, and I’m sure I’ll turn to them often in the future.


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Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.

See the table of contents in the publisher's webpage.