Customer Reviews

Abstract Algebra

5 star:		(25)
4 star:		(14)
3 star:		(3)
2 star:		(2)
1 star:		(4)

 

 

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).

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).

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.

(When I wrote this review D+F was almost one hundred and fifty dollars. It seems to have dropped in price significantly since then)

D+F tries to straddle the line between being a book for advanced undergraduates and a book for graduate students and does a decent job. It is fairly readable, with many excellent exercises and lots of examples. The book also covers all the material in the standard graduate algebra sequence. The section on group theory is particularly good.

I think the biggest problem with D+F is that it is bland. The exposition isn't a joy to read and full of motivation like that of Halmos, Stillwell, or Eisenbud and it isn't full of deep insights like that of MacLane, Lang, or Artin. In addition Category Theory is pushed off to an appendix at the end of the book rather than integrated through the text. Finally the book is expensive and the binding is terrible.

If you want to learn algebra I would recommend purchasing some of these cheaper more focused texts since almost everything in D+F is treated better elsewhere:

Basic Algebra - Mac Lane + Birkhoff - Algebra 3rd Edition

Galois Theory: Stillwell - Elements of Algebra, Artin - Galois Theory

Commutative Algebra: Eisenbud - Commutative Algebra With a View Towards Algebraic Geometry

Homological Algebra: Weibel - An Introduction to Homological Algebra, Mac Lane - Homology

Representation Theory - Fulton + Harris - Representation Theory

If on the other hand you are already fairly comfortable with algebra and are looking for a one volume reference I would just buy Lang. It is less than half the price, more advanced, and has more material.

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...

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.

There are many topics covered in this book, from the beginnings of set theory to the concepts of groups, rings, modules, and fields, and even a bit of representation theory as an added "bonus". This book would probably make a great tool for someone who's had a bit of background in the material and is looking for a reference.

Unfortunately, it also has a few problems. First of all, it most definitely does NOT always choose the most "understandable" approach to teaching concepts, or even in its use of notation. Probably the most stunning example was at one point during the discussion of groups of medium order when the author inexplicably and without explanation chose to use the same variable to represent two completely different things...in the same paragraph! Needless to say, it was a bit confusing the first time through. Also, although the book is largely error-free, there are still a few noticeable mistakes, such as when it claims that closure under addition and multiplication is sufficient for a subset of a ring with identity to be a subring (take the positive integers as a counterexample).

Perhaps the greatest strength of this book is in its large number of problems, often 30 or more for a single section. Unfortunately, this leaves less material to be covered in the chapter itself. I would have liked to see more examples of analysis of all groups of a given order, but those were saved almost entirely as "Exercises for the reader"

This book tries to accomodate both advanced undergraduates and beginning graduates. It would be a good challenge for undergrads to look at, and it covers most of the important topics you would see in a first-year graduate class. It should be readable for graduate students (compared to, say....Lang). Lang, of course covers far more material, but then again,...this is much more readable. One thing which would be a minor complaint is that several central, important topics are relegated to exercises. This is fine for a lot of things you want to introduce, but certain concepts are so fundamental (such as inverse and direct limits, p-adics, trace and norm of field extensions, Hilbert's theorem 90, etc.) that need to be presented within the text. Of course, it could be just my viewpoint that these are such important things in the first place. Overall, it's very readable and I got a lot out of it, even if I don't use it as a regular reference. I can't help but comment on one of the previous reviewers criticism that the authors make a "mistake" in reference to subrings. The mistake is entirely the reviewer's, who failed to read closely. In this book, a "ring" is not required to have identity, so 2Z (or nZ) is in fact a subring of Z, and the claim about subrings is true. The reviewer assumed without reading closely that identity was required. Just felt like vindicating the authors on the point!

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.

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!