Elements of Abstract Algebra

Contents (w.o. subsections):
1. Set Theory
2. Group Theory
3. Field Theory
4. Galois Theory
5. Ring Theory
6. Classical Ideal Theory.

One thing I also liked is that the exercised are scattered throughout the text rather then collected at the chapter ends. You read something and immediately work on a couple (or more) of questions. You understand at the spot rather than waiting the chapter end.


One of the Best Algebra Workbooks in Existence   October 16, 2001
George E. Hrabovsky (Madison, Wisconsin United States)
12 out of 14 found this review helpful

This book is certainly not for everyone. If you prefer a book where you are held by the hand through the material, where you are fed the interpretation, and where all of the work is done for you then do not buy this book. This book is for people who not only want to memorize facts about algebra, but also want to learn to do algebra. The only way to learn to do algebra (or anything else for that matter) is to do it. For example, the first section is (reasonably enough) on sets and has nine subsections. Within these nine sections you are expected to perform nine tasks. This is done in three and a half pages. The section on symmetric groups has ten sections and eighteen tasks in eight pages. This averages to a fraction more than three tasks per page for a 196 page book. This is a lot of problems to work through! It is not so many that the task is impossible in a reasonable period of time. Will you solve every problem the first time? No. Many of these are quite challenging. If you at least study each problem and spend at least five minutes trying to understand it, by the time you are done with the book you will have a good understanding of abstract algebra, and you will be prepared to grapple with more elegant treatments of the subject.


One of the most insightful introductory algebra books   April 3, 2005
Adam Nordloh (USA)
3 out of 3 found this review helpful

I'm a math undergrad, and we're using this as our class text. While some of the criticizms in other reviews are true, Clark's treatment of algebra is thourough, rigourous, and full of many details that other books leave out. While it's true that this is a very concise text, I've found that Elements of Abstract Algebra offers deeper, richer insight into the topics it covers when compared to other intro books.

As an example - cosets. Many other texts completely leave out the fundamental concept of cosets: they are congruence classes modulo a subgroup. In at least three other intro texts I've looked at, the left coset of a subroup was simply defined as gH = {gh | h an elt of H}. While this is true and easier to cope with at first, Clark offers full discussion and suggests where the reader needs to fill in the gaps with proof.

For at least the first two chapters, the reader may want to consider supplementing this book with another, simpler book like Maxfield's "Abstract Algebra and Solutions by Radicals" (another great book). However, any beginner with enough time and discipline will find Clark's book to be a thorough and enlightening introduction.


Ideal for Self-Study.   July 27, 2002
Josh J. Wiley (configuration space)
4 out of 12 found this review helpful

I used this for a first-time self-study of abstract algebra at the undergraduate level. Being an autodidactic math student, I was acclimated to the methods/meaning of theorems and their proofs at what apparently was an unusually early stage (compared to the norm for American undergrads). For this reason, my incredibly positive experience with this book may not carry over for many students learning algebra for the first time. So I'll just say that if you've gained a certain amount of logical intuition (say from a mathematical logic, discrete math, or comp. sci course), have a working intuition of general mathematical problem solving (check Polya's classic if you're in doubt), and have that driving lust to experience the way proofs from "the Book" seem to portray the essential meaning of a theorem in an irreducibly perspicuous fashion, you should find that this text provides you with all the essential materials to fashion for yourself a beautiful and meaningful mathematical experience.
The exercises start (a chapter) at trivial and start to become challenging at a bit past the chapter's midpoint. By this time a working intuition of the basic tools of the theory covered by the chapter is in place, so that the challenge is accompanied by that excitement of using them to create, develop, and eventually establish meaningful conjectures which serve as guideposts to the ultimate solution of the problem. A few ultra-challenging problems in each chapter (where chapters correspond to topics - groups, fields, etc.) did take a number of hours to solve. Problems of this difficulty are fairly few in number, however, and anyone who uses pen/paper should be able to cut my purely-mental problem-solving times down substantially (with some loss in fun). Still, there may be some deadline problems in a classroom environment.
Lack of solutions to the exercises shouldn't be an issue for someone with the aforementioned problem-solving intuition, for they know that:
A. There's more than one way to solve a problem.
B. Your way may not appear in the solution manual, despite being correct.
Therefore:
C. The manual is not going to serve as any better an indicator of
correctness of understanding/proof than will the requirement of consistency with the conjunction of all the preceding (i.e., easier) theorems.
Now:
D. If you've got problem-solving intuition, you can sense when you're right (your brain yells "Eureka!" and everything glimmers with clarity). Re-checking at a later time is helpful, but you'll find your intuition remarkably accurate. Doubting it during solution-process is _very_harmful_ (psychologists and test-taking/problem-solving experts commonly refer to this as 'self-talk'); the "trust now, doubt later" strategy seems best.
So:
E. If you need a solution manual, you'll have to be able to sense how the paradigm solution coheres with yours. If you have this sense, however, you shouldn't need the manual.
Assumably that's why so many books at around this level (and above) lack solutions. Well, that or the publishers seek to drain us poor disciples of all resources via the additional price of a seperate solution manual - but that doesn't apply to Dover books anyway.