Topics in Algebra

Rating:
Summary: Good Introduction, useful for self study
Review: I am an engineer by training and a sales man by profession, with a a strong liking for mathematics.
I found this book to be an very readable introduction to a subject (abstract algebra), I had never been exposed to during my engineering math - other than matirx theory, which was obviously taught extensively.
The proofs are generally easy to understand, but certainly not trivial.
A pleasure to read

Rating:
Summary: The greatest introduction to algebra
Review: I knew before I read Herstein that it was a very famous book known for its exposition and interesting problems. But I had no idea of the reality: it IS amazing! Herstein's approach is to just concentrate on a few basic notions and take it as far as possible before introducing new ideas. This results in very simple-seeming proofs which flow elegantly into the next theorem and proof. Incidentally, Herstein's approach is to also have a bunch of problems that are more meant to be 'tackled rather than solved.' He hopes that by trying to solve hard problems, the reader will come across ideas which are later explained in the book. At that stage, the new ideas are natural. This means these problems are very difficult, and even if you read ahead, they remain difficult. Not to say there aren't some easy ones, but I'd say somewhat less than 50% are difficult. But it's all worth it. I recommend studying out of this book in conjunction with a more standard reference type textbook. Then you get the best of both worlds.

By the way, this book contains an intro to Galois Theory! How many books intended for undergraduates have such topics and such a prestigious reputation?

Rating:
Summary: Good Undergraduate Text
Review: I worked through this book almost twenty years ago. I enjoyed its spare style. I thought it well-crafted. Some of the things I enjoyed about the book were:

1) The three distinct proofs of the Sylow theorems

2) The chapters on linear algebra, in particular where modules over the ring of polynomials (a P.I.D.) were used to derive results on canonical forms and decomposition into cyclic subspaces; this work mimicked what had earlier been done for finitely-generated abelian groups.

3) The elegant discussion of Galois theory.

4) The final chapter on relatively deep results such as Wedderburn's theorem and the sum-of-four-squares (I think).

I am quoting from memory. If I looked at it again I might find more things to admire. There are lacuna, as one reviewer has noted. But in mitigation I should point out that this book is an undergraduate text, and does not aspire to compete with the books of Lang and Jacobson.

Given the present sorry state of American undergraduate education, it is likely that all but the strongest undergraduates will find this book a tough slog.