Customer Reviews

Elements of Abstract Algebra (Dover Books on Mathematics)

5 star:		(7)
4 star:		(2)
3 star:		(2)
2 star:		(0)
1 star:		(0)

 

 

This is a book whose level is between an undergraduate (e.g. Herstein) and a graduate algebra book (e.g. Hungerford,Jacobson). I am a graduate student and I used it for a quick review and i really liked it. It is a little book of 200 pages. One interesting feature is that it first covers field & Galois theory and then ring theory.

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.

I used the previous version of this book while I was a mathematics graduate student at Duke University in 1982. I have never seen a better book for LEARNING field and Galois theory; however, this book is not intended as a reference source. The exercises lead one incrementally through the theory, and this is certainly the best way to learn abstract algebra. I lost my copy of the previous version, but have replaced it with the new one - to have a copy to lend to my own graduate students who want to learn this material.

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.

Since the reviews have been generally positive, I'll start with the major negative. Clark does a poor job of motivating the material being developed. As a reader with no background in modern algebra, I found the group theory chapter tedious and uninteresting. Just because you can begin with a set of definitions and use them to prove very complicated theorems doesn't mean doing so is worthwhile. It wasn't until I read the fourth chapter on Galois Theory that everything clicked and I realized the importance of seemingly arbitrary definitions and correspondingly ponderous theorems. But even then I had to do considerable introspection. The proof that polynomials are solvable by radicals iff the Galois group of transformations is solvable is presented as just another theorem, whereas that proof is the principal purpose of most of the book to that point. I basically had to figure out Galois's original idea for myself and then go back and reread Clark's chapters 2-4 for the complete analysis. To be fair, this book has an introduction that sort of hints at Galois's idea, but I feel it is very poorly done. Perhaps a more thorough, more motivational introduction would make this a 5-star book.

Sometimes Clark appears needlessly complex. In one part, he defines the normalizer of a subgroup as the group of all elements in which the subgroup is normal. Then he proves, in a bizarre and tedious way, that the normalizer is the largest group in which the subgroup is normal. While I'm not a mathematician, it seems to me that this is obviously true by definition.

On the other hand, you can learn a lot from this book quickly precisely because of its compactness. I am fond of concise writing, but the whole purpose for a book is to guide the reader's thought. I almost recommend beginning this book with chapter 4 unless you have already expended considerable thought on equations.

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.

I wanted to study Galois Theory to understand why the quintic is not solvable in radicals. I did some search on the net and ran into this book. My math background is in probability and analysis. With my background and interest this book I feel this book is perfect. It is not too difficult, plenty of exercises and I can follow the development; also I do not feel I am being talked down to by the author. I will have a good understanding of Galois and related theories after putting in the time and effort with this book.

I recommend this book for all who are taking undergraduate Abstract Algebra. The book gives clear explanation of most of the concepts taught in class. Another great thing about this book is that it includes definitions and explanations of terms that are usually not discussed in class. This is a must have for math majors!

It's all already been said: this is a great book if you're looking to thoroughly learn the material yourself, rather than be given proofs. It forces you to prove fundamental ideas for the development of the theory, providing definitions and guidance. It's absolutely full of relevant exercises.
I really liked the style of the book. It felt more like a mathematical discussion rather than a lesson. This was partly due to the many exercises immediately following each definition or theorems, rather than being chunked all at the end of the chapter. I didn't feel like I was being talked down to, which I think is deserved for getting through the exercises myself.
However, it was exactly for these reasons that my boyfriend didn't like the book, so be aware that it just might not be your cup of tea. It's worth trying out, though, considering the low price.

This book itemizes the related theorems and covers most major results. It would be better if some definitions have some more detailed examples.

It is inappropriate for self-study .