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Abstract Algebra (Graduate Texts in Mathematics, 242) 2nd Edition

4.0 4.0 out of 5 stars 10 ratings
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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.

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  1. ISBN-10
    0387715673
  2. ISBN-13
    978-0387715674
  3. Edition
    2nd
  4. Publisher
    Springer
  5. Publication date
    July 24, 2007
  6. Part of series
    Graduate Texts in Mathematics
  7. Language
    English
  8. Dimensions
    6.32 x 1.38 x 9.51 inches
  9. Print length
    686 pages
  10. See all details
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Editorial Reviews

Review

From the reviews of the second edition:

"This book certainly has more than enough material for a two-semester algebra course for first-year graduate students. … The balance among various topics will please some instructors more than others." (Miklós Bóna, MathDL, September, 2007)

"The book under review is the second edition of the author’s sweeping text originally titled ‘Algebra’ … . All together, the second edition of P. A. Grillet’s standard primer is a masterly, clearly written, comprehensive and self-contained algebra text for graduate students, with a plentiful supply of additional advanced material … . Moreover, the book under review, in its didactically oriented disposition, represents an excellent source for teachers and instructors of graduate courses in abstract algebra as well." (Werner Kleinert, Zentralblatt MATH, Vol. 1122 (24), 2007)

From the Back Cover

"Abstract Algebra" is a clearly written, self-contained basic algebra text for graduate students, with a generous amount of additional material that suggests the scope of contemporary algebra. The first chapters blend standard contents with a careful introduction to proofs with arrows. The last chapters, on universal algebras and categories, including tripleability, give valuable general views of algebra. There are over 1400 exercises, at varying degrees of difficulty.

For the new edition, the author has completely rewritten the entire text, streamlining the first chapters for rapid access to Galois theory, removing some material, and adding introductions to Groebner bases, Ext and Tor, and other topics.

From a review of the First Edition:

...combines an exceptionally accessible discussion of the basic material with a just as thorough and well-organized treatment of the many additional (advanced) topics included.... represents an outstanding introduction to modern abstract algebra as a whole, with many unique features. It captivates the reader by its remarkable diversity, comprehensiveness, elegant succinctness, and coherence.

- Werner Kleinert, Zentralblatt

Product details

  • Publisher ‏ : ‎ Springer; 2nd edition (July 24, 2007)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 686 pages
  • ISBN-10 ‏ : ‎ 0387715673
  • ISBN-13 ‏ : ‎ 978-0387715674
  • Item Weight ‏ : ‎ 5.51 pounds
  • Dimensions ‏ : ‎ 6.32 x 1.38 x 9.51 inches
  • Customer Reviews:
    4.0 4.0 out of 5 stars 10 ratings

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Pierre A. Grillet

Pierre A. Grillet

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Customer reviews

4 out of 5 stars
4 out of 5
10 global ratings

Top reviews from the United States

Jason Schorn
5.0 out of 5 stars Full of Insight
Reviewed in the United States on February 14, 2004
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!
18 people found this helpful
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Matthew Byrne
3.0 out of 5 stars Good but skims a lot.
Reviewed in the United States on December 4, 2013
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Great book very well thought out and well reasoned proofs. I wouldn't give it four stars as it seems to be quite advanced and "skims" a lot. Therefore I wouldn't say it's a graduate text. I have read many other books on Abstract Algebra that conveyed the same concepts but they were more gentle.

Other than that it is a solid book that would make an excellent resource for anyone studying this area of mathematics.
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Mr. Bubbles
3.0 out of 5 stars has great potential, but could use a second edition.
Reviewed in the United States on November 11, 2006
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.
8 people found this helpful
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sowmya
1.0 out of 5 stars Not recommended
Reviewed in the United States on December 17, 2002
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.
4 people found this helpful
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