Advanced Modern Algebra

Rotman's book has just about anything you'd want out of the sort of fat algebra bibles that have become popular recently. It's comparable to Dummit and Foote, though the second is written at a somewhat higher level and provides a unified treatment of commutative and non-commutative rings. Still, I think that Rotman is a bit wordy and rather than clarify, his lengthy proofs tend to obfuscate. (His proof of the correspondence theorem is particularly guilty of this.) I think the gold standard as far as presenting clear and concise proofs in algebra is concerned is the book by Hungerford. I used Hungerford in conjunction with this book. First I'd read Hungerford to get a crisp, clean and concise version of the theory and then I'd read the relevant parts of Rotman for the extra examples and intuition. Another thing I don't like about this book is the physical size. I think the book could have shed some weight by using a smaller font and better typesetting without sacrificing any of the content.

I can't believe someone actually said the problems in this book are easy! That's outrageous! This book is rigorous and is designed for graduate Abstract Algebra. The first few chapters may seen easy because they are a review of some topics down in undergraduate Abstract Algebra. I think the author did a very good job of reviewing first and then getting into the deep stuff to give all the students a fair chance. Not everyone is brilliant! I highly recommend this book for Graduate Abstract Algebra.

This is a tough book to review, because it is not clear who the real audience is supposed to be. The author says that it is aimed at first-year graduate students, with a bunch of extra material that can be referred back to during the second year and beyond. The earlier chapters also include efficient reviews (with sketched proofs) of material that should be familiar to those who have taken undergraduate algebra.This characterization is debatable. Based on my experience reading most of the first six chapters (the first 400 out of about 1000 pages), I would say that the level of sophistication is roughly that of Dummit and Foote's "Abstract Algebra", which is usually considered an undergraduate book. D & F can sometimes be harder to read, and that is in part because Rotman's exposition is better (in my opinion), but also because D & F introduce more difficult material earlier. Whether D & F's approach is better is questionable; I find Rotman to be a much smoother read, but the organization is quite different -- for example, one does not encounter noncommutative rings until deep into the book, whereas Dummit and Foote introduce them immediately upon defining rings. On the other hand, early in the coverage of D & F's chapter on rings, one has to digest Zorn's Lemma and its applications almost from the beginning, whereas Rotman (I think wisely) pushes this back into a later section. In general, D & F introduce a lot of hairy examples that by themselves require a lot of effort to digest (thereby impeding the reader's progress through the core material), whereas Rotman's examples tend to be straightforward, at least as new concepts are being presented.So, overall, the exposition flows more smoothly in Rotman's book, and the reader can cover the basics more quickly with less time spent on tangential examples and early generalizations. Also, Rotman's proofs are usually much cleaner and the overall style is very nice. It's more pleasant to read than Dummit and Foote. But this comes at a cost: Dummit and Foote do cover more material, and generalize at an earlier stage, than Rotman does.But my biggest gripe concerns the exercises. Put simply, Rotman's are far too easy for what is being pitched as a graduate course. In fact, they are in general far easier than the homework problems I sweated through when I took honors undergraduate algebra. They're barely adequate to convince the reader that he has a basic grasp on the material, and there are almost no hard ones, let alone really tough, thought-provoking open-ended problems like one encounters in Herstein's "Topics in Algebra" (an undergraduate book). There are certainly no exercises in Rotman's book that would be of any use for a graduate student preparing for qualifying exams. They're not even much of a workout for a decent (honors student) undergraduate.So, what is this book good for? I think it's great for reading material that is usually harder to understand elsewhere. Rotman has a real k

To begin with, don't let the title scare you. After having read through Rotman's book I am suprised that this text had not crossed my path earlier. It is a wonderful book and must have for any inspiring Algebraist. Moreover, I am quite shocked that the larger universities have not adopted this book.(a) This book could quite easily be used as the standard third/fourth year undergraduate introduction to Abstract Algebra. In particular, the first four chapters provide a solid foundation for a moderate paced one semester course at which point the instructor has many different options for additional topics based on the performance of his/her class. (b) Those students that move on to the graduate level, and obviously to a university using this book, would both be familiar with the temperment and flow of the author as well as devoid of the requirement of having to purchase another expensive Mathematics text. For example, my undergraduate Algebra text was Hungerford's and post completion the logical step, being familiar with his style, was to purchase Hungerford's graduate text. For those not familiar, let me tell you there is a night and day difference with repsect to how the material is presented. (c) The remaining 7 chapters take the willing student on a pleasant tour of ring/module theory, some advanced group theory (for the inspiring group theorist I highly recommend the authors graduate text "Group Theory"), algebras(linear included), Homology(some cohomology) and finally some algebraic number theoretic concept under the heading of Commutative Rings III.(d) Lastly, Rotamn does not get needlessly bogged down in any one section of the book. The flow is smooth, to the point with precise definitions, examples, and ample exercises.I have only two negative remarks: one, the failure to include more aspects of field/Galois theory. This may be due to the author already having published a book entitled "Galois Theory". Two, the failure to devote an entire section to Finite Fileds and possibly some its applications. But this failure is minimal since, at present, the majority of Algebra texts, fail to adequately introduce and motivate Finite Fields.

I previously purchased Rotman's First Course in Abstract Algebra, and fell in love with it. So when I saw he a Second Abstract Algebra book, I had to have it. I am currently taking a Graduate Level Modern Algebra course, and I find this book to be a great help in my Studies. I wouldn't be as interested in Modern Algebra as I am now if it weren't for this book. I love this book and I would reccomend it to anyone who is interested in Modern Algebra, or taking a course in Modern or Abstract Algebra.