Algebra : A Graduate Course (Mathematics)

Rating:
Summary: One of the best math books I've ever read
Review: If you are looking for a great first book on abstract algebra, this is it! Dr. Isaacs has written a self-contained work that covers the basics of the subject in an easy to read manner. This book assumes that the reader has no previous knowledge of modern [abstract] algebra, though some mathematical maturity is required. It also avoids the twin pitfalls of mathematical writing: "Theorem, proof, theorem, proof,...", and "The details are left to the reader."

Rating:
Summary: A great first book about Abstract Algebra
Review: If you are looking for a great first book on abstract algebra, this is it! Dr. Isaacs has written a self-contained work that covers the basics of the subject in an easy to read manner. This book assumes that the reader has no previous knowledge of modern [abstract] algebra, though some mathematical maturity is required. It also avoids the twin pitfalls of mathematical writing: "Theorem, proof, theorem, proof,...", and "The details are left to the reader."

Rating:
Summary: One of the best math books I've ever read
Review: Isaacs' algebra text is probably the best math book I've encountered so far as an undergraduate student for several reasons.

First, the structure of the book is unique- most introductory algebra books tend to cover groups, rings, and fields in that order. More mathematically mature students, though, can gain a greater appreciation for rings by first understanding modules. Most texts tend to introduce rings first, because the classic examples of rings are easy to understand, and then generalize to modules. Isaacs instead builds upon the composition structures of groups to introduce the topic of X-groups (this is the only introductory graduate text that covers this extensively), so that modules and rings are not only presented at the same time, but in such a way that the reader can see the interplay between the two. This presentation also makes it easier to discuss the Jacobson radical and by the time the Wedderburn-Artin theorems are presented, the reader is familiar enough with the necessary elements of the proof that it actually becomes easy.

Another reason this book is good is because Isaacs includes difficult topics not generally covered in an introductory text, but in a way that they seem to be just a simple extension of the more basic material. For example, at the end of the noncommutative section (the first half of the book), Isaacs proves the algebraic foundation of character theory using the Wedderburn-Artin theorems, showing the module presentation of a representation as well as the classic homomorphism presentation. He then proves the basic results about characters, giving a very powerful tool to analyze the structure of a group.

In a more applied vein, Isaacs proves the steps used in the Berlekamp algorithm in the finite fields chapter, which not only allows the reader to gain experience using the generalized Chinese Remainder Theorem but also to apply it to the study of fields. After covering integrality, Isaacs explains the role of rational integers in character theory and applies it to prove Burnside's celebrated solvability proof, whose statement about groups seems to have nothing to do with integrality, or even noetherian rings for that matter.

While Isaacs covers other advanced topics (for example, Transfer theory in the study of groups, or the Schraier-Artin theorem), the text is excellent because he proves the basic results so clearly. While he doesn't talk about the geometric significance of groups that much, he does talk about groups from a stabilizer-orbit perspective that makes further study of symmetries a lot easier.

The proofs of the Fundamental Theorem of Galois Theory, Galois' proof of solvability, the Principal Ideal Theorem, and a stronger form of Sylow's theorem are particularly elegant, along with the chapter on solvable and nilpotent groups. What makes the book far superior to others, though, is the problems. If you can understand the hard proofs of this book, you should be able to do the problems in easier books (Dummit and Foote, Hungerford) pretty easily. Be warned- the problems are not there to have you "fill in the details" Isaacs left out (because his proofs generally don't leave even minute details out) or to get practice, but to actually prove new results. For example, important topics such as metabelian groups, supersolvability, and the structure of a field with an abelian Galois group are presented as problems.

In sum, anyone who wants to appreciate the beauty of algebra and understand more than just the basic concepts should learn it from Isaacs' book. While it is self-contained, one may want to study Herstein's book first and do some problems so that this book doesn't seem as intimidating. After studying this, you should be prepared to answer any basic algebra question on any prelim exam in the country and be sufficiently prepared to tackle more advanced branches of algebra.