Abstract Algebra (Graduate Texts in Mathematics)

About the first edition:

“The text is geared to the needs of the beginning graduate student, covering with complete, well-written proofs the usual major branches of groups, rings, fields, and modules...[n]one of the material one expects in a book like this is missing, and the level of detail is appropriate for its intended audience.” (Alberto Delgado, MathSciNet)

“This text promotes the conceptual understanding of algebra as a whole, and that with great methodological mastery. Although the presentation is predominantly abstract...it nevertheless features a careful selection of important examples, together with a remarkably detailed and strategically skillful elaboration of the more sophisticated, abstract theories.” (Werner Kleinert, Zentralblatt)

For the new edition, the author has completely rewritten the text, reorganized many of the sections, and even cut or shortened material which is no longer essential. He has added a chapter on Ext and Tor, as well as a bit of topology.

On some subjects, it is more of a survey of topics (such as category theory, universal algebras, exterior products), the point of which only becomes clear after one needs to use these tools in other areas.

If you are new to the subject, however, the texts by Dummit and Foote or Fraleigh, e.g., are more appropriate.


another good algebra text   July 29, 1999
6 out of 7 found this review helpful

This text is designed for beginning graduate students. The book includes all the basic parts of algebra any mathematician should know. The presentation and proofs are clear and easy to follow. People with no prior exposure to abstract algebra might have problems learning algebra from this book as quite a few important theorems and results are left with no proof. Instructor can easily supplement those missing proofs if he/she thinks that's appropriate. Overall, it's an excellent reference book for researchers, but only a good textbook for students.


has great potential, but could use a second edition.   November 12, 2006
Mr. Bubbles (NY, USA)
The title of the review pretty much says it all. For a first edition, this isn't bad. I've taken copious notes from this book, and I've learned quite a bit. It's much more comprehensive than most other algebra books on the market, to be sure. I'd be hard pressed to find another introductory algebra book that actually makes it to the Adjoint Functor theorem, tripleability, and Birkhoff's characterization of varieties. However, the book also has its flaws. The most annoying one is probably Lemma 10.6.8 on page 331, which says that a finite dimensional division algebra over an algebraically closed field has dimension 1. This is clearly false. (Consider the quaternions and the complex numbers). The error in the proof is that he assumes commutativity (probably the easiest blunder to make in algebra, so it's a minor offense).Thankfully, no subsequent use of this "lemma" is made in the book (making me wonder why he's choosing to call it a lemma). In some places, Grillet doesn't really have the slickest proof on the market, which would be nice if he's trying to be comprehensive (the book wouldn't be quite as much of a wrist cracker). For instance Jacobson's proof of the simplicity of PSL is much slicker and isolates the hypotheses (dim V >2 or |K|>3) The chapter dependency chart is useful, although he breaks the logic somewhat by using fields in the chapter on group theory.Chapter notes at the end of each chapter would be very useful.
All of these problems can be easily fixed in a second edition. Another idea would be for the author to maintain some sort of errata page (John Lee does this for his books on manifolds). For the most part, the book has great potential because it's got a nice, ambitious logical structure that you won't find elsewhere. I'd rather see someone go out on a limb and try to write a comprehensive, up to date, state of the art algebra book than simply rewrite an existing book focusing only on classical algebra.
For professors who are thinking of using this book for their algebra classes, I'd suggest going through it yourself before the term starts (give yourself a few months) so that you can tweak it a little. That way if you use the book for you class you can catch any mistakes.


Not recommended   December 17, 2002
sowmya (Sunnyvale,CA,USA)
4 out of 11 found this review helpful

A bad textbook I have read for algebra.The author seems to be complicating every concept thats simple to understand.
Also most of the proofs given are not straight forward and forces you to refer to some other books for a clear understanding and definitions of the same concepts given in the book.
Not at all recommended.
Better buy a Fraleigh or Herstein or Thomas Hungerford even if your teacher recommends this one.