Customer Reviews

Abstract Algebra (Graduate Texts in Mathematics)

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This book does not complicate every concept, on the contrary, and for example, this is the only algebra text I am currently aware of that actually provides insight into the 'mechanics' of morphisms between various types of sets. To be precise, in section 1.6 (The Isomorphism Theorems), Grillet, introduces and describes both factoring through a domain and codomain. For the student, this provides insight into what is happening 'behind-the-scenes' in the homomorphism theorem and,in turn, acts as a vehicle that enables the student to fully understand and appreciate the isomorphism theorems. Of course one must take in to account the ability of the student using this text and, given this, I would say Grilet's text requires nothing more than an elementary introduction to the basic algebraic structures. This text is not overwhelmingly wordy like that of Dummit and Foote or baby Hungerford, nor is it stale and lifeless like Lang's. Instead, it is well written, definitely insightful, covers all the material needed at the begining graduate level and this text can be used, like Rotman and Lang's text, as a reference. If this book seems daunting based on the size, then I would recommend either just dealing with it or using Martin Isaacs or Papa Hungerford's since both are complete, challenging and manageable. Enjoy!

Having used this book in a graduate algebra course, I feel that it is an excellent text for those who have already had a rigorous introductory exposure to modern algebra (say, via Dummit and Foote or Fraleigh). Grillet writes clearly and concisely and leaves several challenging (but doable) proofs to the reader. His wry sense of humor is also reflected in his writing.

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.

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.

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.

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.