Abstract Algebra

This is the only book I bought as an undergraduate that I still look at today. All my other undergrad texts are either stored away somewhere or gather dust on my bookshelf. The reason is simple: Dummit and Foote has stocked in one book almost all the basic algebra that is required for my study of 3-manifold theory. I suspect this is true of other fields also. By "basic algebra" I mean the key ideas and examples that are used in many different areas of mathematics.

Just recently, I needed to pick up some algebraic geometry in order to understand SL(2, C) character varieties. As usual, I went to my Dummit and Foote and found what I needed (for the most part). And also as usual, I will need to supplement that knowledge with some more advanced books.

A couple things about this book annoy me though: 1) the price -- however, I have certainly gotten my money's worth out of it over the years, so I can't really complain 2) Initially when I first got the book, the wealth of material in the book appeared intimidating and esoteric to me; however, nowadays I would say there isn't *enough* in this book. Oftentimes it seems that I get just a taste before the discussion of a topic ends. On the other hand, I am realistic, so I realize that this book is not meant to be encyclopedic but to introduce the reader to the more advanced topics.

I've yet to see another book that carries all the topics of this one, and remains fairly reader-friendly (as this one does).


The book after Herstein   December 12, 2004
Michael B Williams (Brooklyn, NY)
26 out of 27 found this review helpful

I think I would only recommend this book to someone who has already had some exposure to algebra (or one especially gifted in mathematics). The beginning of the book is not too bad, but towards the end of Part I the pace quickens quite a bit. If you are willing to read over the text many times, and do all of the non-trivial exercises (there is an impressive olla podrida of algebra in them, most of which are the beginnings of some very deep ideas), then it should be a very rewarding experience. Namely because this is one of the most readable textbooks which covers everything from groups, rings, and fields to homological algebra and algebraic geometry. It is very rare to see this much material covered in one book, and for it to remain so structured (Rotman is an example of a book that covers a lot of material, but loses its structure somewhere).


slick development of algebraic concepts   December 14, 2001
Marisa Debowsky (New York, NY)
9 out of 9 found this review helpful

This book arose from the lecture notes of Dave Dummit and Richard Foote, both at the University of Vermont. I first encountered this book in Dummit's own graduate algebra I-II class and was swept away by the clarity in contrast with my previous classes. Both authors are excellent teachers, and their text is equally good. Really slick development of group and ring theory, in an intuitive manner, constantly working through examples with symmetric and dihedral groups. Also includes high-level algebra suited for topics courses. I highly recommend this text.


More than complete if (perhaps) not introductory   December 30, 2002
Josh J. Wiley (configuration space)
24 out of 28 found this review helpful

For a number of reasons this may not be the best book for undergraduate self-study:
1. No answers to problems (though I think this should be much less a problem for anyone doing abstract algebra at any level, I'll stay off the soapbox)
2. This book contains a lot of information beyond the basic (undergraduate) essentials, and as this extra information is quite densely packed into each part of the book, it might be tough to pick out the main points
3. The exercises stay (for the most part) at a relatively uniform/low level of difficulty, but the proof/calculation ratio is kind of high (still resisting soapbox-related urges ...)

I'm sure there are others; many have been mentioned in previous reviews.

For graduate-level self-study, however, this book is a dream. As mentioned above, it is overflowing with information at every turn, which keeps the stuff that's review interesting and the stuff that's new accessible (at this level students should have the toolbox to deal with examples and such that draw from analysis, topology, or what-have-you). It has chapters on commutative algebra, homology theory, and representation theory (of finite groups), and appendices on Zorn's lemma and category theory. The conversational style isn't distracting (a big issue for me), possibly because of the exceptional organization for a book covering so much. Finally, the authors have succeeded tremendously in presenting everything with a view toward its ultimate use by the reader further along into "the great mathematical beyond" (I apologize for using this phrase).

One complaint: I can't seem to find a bibliography...


The book for advanced undergraduate work   September 9, 2000
David Rudel (Charlottesville, VA)
7 out of 8 found this review helpful

This is _THE_ book to pick for advanced undergraduates, especially for those who wish to learn on their own. This is also the best "bridge" book between undergraduate concepts and graduate concepts.

The biggest plus the book has is the simple fact that it is willing to take some time to explain, rather than state, applications, concepts, and theorems. It has an exceptional rendering of Linear Algebra.