Customer Reviews

Algebra: A Graduate Course (Mathematics)

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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.

I love this book; its shortcomings do not even tempt me to rate it less than 5 stars. However, it is not perfect, and you should consider its shortcomings when deciding whether or not to use it for a course or add it to your collection.

The book is exceptionally clear, even though it is dense. However, this book is not appropriate for a first course in abstract algebra: even if you have a strong mathematical background, it definitely requires prior experience with abstract algebra. The notation is a bit eccentric at times, but it is consistent, and once you get used to it, the book reads very easily. It is excellent for self-study.

The book is organized so that advanced chapters on specific topics can be read on their own, making this book an outstanding reference, and also making it easier to design a course using this book as a text. In the occasional cases when earlier material is required, the indexing is excellent, and the definitions are clear and concise.

Exercises are very illuminating, and diverse in difficulty level.

At times, Isaac provides tedious (and sometimes confusing) proofs of results which are intuitively obvious and in my opinion are best left to the reader. Isaac's style of proof seems to be to include every detail, even when leaving certain details to the reader would actually make the proof easier to comprehend. This space in the book, in my opinion, should be occupied instead with concrete examples, which leads into my largest complaint:

This book is almost completely devoid of concrete examples. Numerous results are proven before a single example is given, and in some cases, no examples are ever given--the reader is left to construct such examples on her own, or find them in another book. Chapters that provide more concrete examples, such as the one on permutation groups, are strictly optional and are not well-integrated into the text. Working the exercises provides some much-needed examples, but this is still a weak point of the text. This also relates to the other weak point of this text--connections to other areas of mathematics. This book is clearly written by an algebraist who sees the inherent beauty of the subject, and he does an excellent job of communicating this beauty through his writing. However, the connections to other branches of mathematics simply aren't in the book.

This book is complemented by books like Lang's Algebra at a more advanced level, or Dummitt and Foote at a more elementary level. In some ways, this book is the exact opposite of Lang's: Isaacs' proofs are detailed, expanded, but tedious, and Isaacs provides few concrete examples. Lang's proofs are sparse or not present, yet Lang provides numerous examples and countless connections to other branches of mathematics.

Lastly, the of this price is absolutely obscene. Although the book is excellent and certainly worth the money, the binding isn't: it completely fell apart and needed to be glued back together after only moderate use. This book held up less well than most cheap paperbacks. For $157, this lack of quality is downright criminal.

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."