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ByA customeron July 11, 2002

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.

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.

2 people found this helpful

ByRyan Malloyon January 2, 2004

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.

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.

ByRyan Malloyon January 2, 2004

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.

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.

ByA customeron July 11, 2002

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.

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.

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ByJ. Wileyon July 26, 2002

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.

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.

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ByGeorge E. Hrabovskyon October 16, 2001

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.

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ByDavid B. Masseyon August 13, 2001

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.

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ByStefanon March 18, 2016

Excellent book.The problem based approach is definitely most suitable for my learning of this topic.

I just wish there was a solution manual.

I just wish there was a solution manual.

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ByA MATH NERDon March 5, 2000

Or it's not, because she never warned you about how some Dover books are so bad that they shouldn't be in print. But such a warning is necessary. When I first discovered Dover math books, I thought the concept of very inexpensive math books was to good to be true-and it was. I bought this book as a naive teenager, and I really think it's badness set me back several years, in terms of learning abstract algebra. If I had instead bought a good book like Birkhoff and Maclane... There are good Dover books though, and I have done my best to review them(so click on my name if you want to see them).

So, why do I hate this book so much? A very small percentage of it is proofs or discussions of proofs or result-the ratio of these to problems is even lower than in Atiyah and Mcdonald. Not that the book doesn't prove a lot-it does(though not well), though a few important results are pushed to the exercises, without any helpful hints. The big problem is that it seems like Professor Clark wanted to see how few words he could condense his text into, and as such the text is nearly unreadable, absolutely impossible to learn from. The exercises are pretty good though, and if you just want lots of exercises, maybe this book is useful as a supplement. The content, what little of it there is, is absolutely awful. Many of the proofs given are needlessly complicated, in particular this is the case in the important topic of Galois theory. All books written on the subject since 1940 have copied Emil Artin's treatment(available in one of the good Dover books), except this one I guess, which is just weird and ugly.

The one good thing one could say about this book is that it does treat in complete detail all of the classical applications, of constructability and solutions of cubics, quartics, and quintics. In particular, there are far more details on constructible numbers than in other books I've looked at. However, these things do not make up for the many fundamental flaws of this book. Stay away!

So, why do I hate this book so much? A very small percentage of it is proofs or discussions of proofs or result-the ratio of these to problems is even lower than in Atiyah and Mcdonald. Not that the book doesn't prove a lot-it does(though not well), though a few important results are pushed to the exercises, without any helpful hints. The big problem is that it seems like Professor Clark wanted to see how few words he could condense his text into, and as such the text is nearly unreadable, absolutely impossible to learn from. The exercises are pretty good though, and if you just want lots of exercises, maybe this book is useful as a supplement. The content, what little of it there is, is absolutely awful. Many of the proofs given are needlessly complicated, in particular this is the case in the important topic of Galois theory. All books written on the subject since 1940 have copied Emil Artin's treatment(available in one of the good Dover books), except this one I guess, which is just weird and ugly.

The one good thing one could say about this book is that it does treat in complete detail all of the classical applications, of constructability and solutions of cubics, quartics, and quintics. In particular, there are far more details on constructible numbers than in other books I've looked at. However, these things do not make up for the many fundamental flaws of this book. Stay away!

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ByA MATH NERDon February 7, 2000

Back in the days of my youth, I was interested in learning about abstract algebra, and I saw this book at the local Barnes and Noble, for the usual reasonable Dover price. This was one of the first Dovers I had ever bought, so I didn't yet know that many of them were just plain awful, including this one. There are several fundamental problems with this book. First, most of the book is problems, and there is almost no discussion between proofs, and only very concise proofs. Maybe if you just want a good source of abstract algebra problems this book will be ok for you. If this book had proofs 'from the book', that is to say, they had the ideal proofs which make everything seem clear, maybe this approach would have worked. But many of the proofs are unbelievably clumsy compared to what I have seen in other texts.(At least for Galois theory, everything leading up to it, and its applications, an example of a book with optimal proofs is Artin's Galois Theory-and by the way, you don't have to be an eighth grader to read that book.) The one thing that can be said in favor of this book is that is that is does present in full detail all the classical applications of abstract algebra-the cubic, quartic, quintic, and constructible numbers-which are applied to angle trisection and polygon constructability. Moreover, the last chapter has the worst introduction to Dedekind domains I've ever seen.

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