Most helpful critical review
2 of 2 people found the following review helpful
Extremely compact, not enough discussion
on 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.