Rating: 
Summary: This book is clearly written and quite comprehensible.
Review: Dummit and Foote's Abstract Algebra is an excellent resource for any serious student of mathematics. While it assumes that the reader has some previous knowledge of the material, it rigorously builds the entire subject of modern algebra from first principles. Each section is clear and well written, with detailed examples and related illustrations. Of all the standard algebra texts I know, it most effectively combines clarity with technical rigor. All in all, it is a superb text for an advanced undergraduate or a blossoming graduate student. (In response to another review, the exercises have no printed solutions, but it would be ludicrous for a text written at this level to have them.)
Rating:
Summary: More than complete if (perhaps) not introductory
Review: 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...
Rating:
Summary: Excellent book
Review: I am surprised that this book has not got the 5 stars. It is very suitable for advanced undergraduates/first-year graduates. The book is full of examples; and the proofs are amazingly clear and succinct. The book introduces new concepts in the excercises long before the student encounters them in the sections.
This is a beautiful way to teach mathemtatics,--and indeed to learn it. The book is replete with examples that connect concepts from toplogy and real analysis with Algebra.
This book definitely deserves the 5 STARS.
Rating:
Summary: The book after Herstein
Review: 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).
Rating:
Summary: Comprehensive book
Review: I'm a graduate student in math. We used this book for the basic year-long abstract algebra sequence: group theory, chapters 1-4 and some of chapter 5; ring/field/galois theory chapters 7-9, 13-14. Some of my fellow students took a module theory course which was at least partially based off chapters 10 and (I think) 11. I'm sure more advanced courses could easily be based off chapters 15-end. Considering the cost of university books, I consider it very nice to buy one book for essentially 3+ courses. The exercises in some sections are very diverse. My group theory professor made us do a huge number of them, and now I am amazed at how often I see questions similar to those from Dummit-Foote show up on past qualifier exams from many different universities. Regarding lack of answers in the back...well, you shouldn't need too many, and if you get really stuck, that's what the professor is for. And if you're learning it on your own then I'm thinking you should be brainy enough not to need answers! The text itself is very readable and complete. I don't think I'd recommend this as an undergrad textbook, although I've no doubt that there are some clever undergrads who could learn from it. I used Herstein's "Topics in Algebra" for my intro-to-abstract course as an undergrad. Herstein is designed to be introductory in nature, though still a wonderful book, while DF is more encyclopedic. I do have one complaint though: the binding in DF started to crack and pages started to fall out near the end of its first semester of use. It did see some moderate backpack use but not too much, certainly less than many other books I've had in the past. So take care of it!
Rating: 
Summary: This book is rigorous, dense, and lacks answers to exercises
Review: It is a strong, rigorous introduction to abstract algebra, but I find it too intense for most undergraduates to fully grasp without studying for extraordinary amounts of time. None of the exercises have answers provided. so it isn't very useful for self-study. It is abstract in its very quintessence: it gives no examples of applications of the theorems. If you don't have a significant background in mathematical proofs and abstract logic, choose a more accessible text. But, if you can handle the level of rigor and abstraction, the text is very rewarding.
Rating:
Summary: You'll get a lot of mileage out of it
Review: Most of the reviews have been positive, and basically explain the strengths of the book, but I thought some would appreciate hearing what someone, like me, who has gone through most of the material in the book over the last three and half years, would say. 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).
Rating:
Summary: Very good introduction to abstract algebra for self study
Review: This book is very clearly written, and rather comprehensive. It does have a few errors, as has been noted by other reviewers, but I think its strong point lies in the many illustrative examples which can be found throughout the text. I would say that it is strongest when describing group theory and galois theory, but it certainly is good in other areas as well. The many examples and excersizes make it ideal for self-study (I speak from my own experience here...)
Rating:
Summary: If a man's wit be wandering, let him study the mathematics
Review: This is a most excellent text although I'm partial to the impressionists.
Rating:
Summary: Overall, one of the best algebra books out there!
Review: Well, I feel that I should add to the long list of affirmative reviews. Why, I don't know? If you want a book that will take you from the basics of Abstract algebra to the advanced level, then buy this book, you won't regret it. It is a little more terse than, say, J. Gallian's, but I think that that problem can be overcome without too much difficulty. It doesn't have the answers in the back, but if this is your first serious proof-based math course, then you need to start breaking that habit anyway. (Most of the answers one finds in advanced math books are usually the solutions to the hardest approach to solving the problem.) If you buy this book, you won't have to buy another one for at least a year to a year and a half.
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