Abstract Algebra, 3rd Edition

Although the several texts I have seen are useful in their own right, I don't believe there's a better text for beginners (or, perhaps, to strengthen shady concepts for further courses) on the subject. Herstein presents concrete examples before proving abstract concepts (something students who have only had courses on the several calculus, discrete math, probability, and matrix theory will find invaluable).

The text is clear and concise. The length is short without omitting any pertinent ideas (other books tend to spend a wealth of pages on anomalies -- which can be good...but then we could really make volumes on the subject). The book starts with a basic (but complete) introduction to sets, groups, symmetric groups, rings, fields and ends with some special topics (simplicity of A4, finite fields existence and uniqueness, cyclotomic polynomials, Liouville's criterion, irrationality of pi). As with most texts on the subject, there are no solutions provided.

The tone taken in the work caters to the student who has not had a course in abstract theory and proofs (ie- courses in analysis or topology ; a number theory course, in my experience, is not rigorous enough for the average student supply the necessary background).

For more difficult and robust presentations, look towards Artin's "Algebra" or Hungerford's "Algebra" (the latter being part of the Graduate Texts series).

For a more application (example-based) text (usually simpler for the less advanced student), look towards Gallian "Contemporary Abstract Algebra." This text is one of the rare occasions where the odd-numbered problems have solutions (but if that's all you're looking for, go to some of the Schaum's series). It is a bit basic (spending most of its examples on more familiar concepts), but also hits on some historical notes and examples that are good conversation pieces (something more mathematicians could use).

Also, as a sidenote, the editions have not changed the content at all. I would suggest getting an older edition...


Best at what it is   September 23, 2002
Cletus Bojangles (Atlanta, GA)
17 out of 19 found this review helpful

(I am writing about the 2nd edition, which I used as an undergraduate.)

This book is intended for a one semester senior-level honors course at a reasonably good undergraduate institution, for which it is perfect. Students who are less interested in pure mathematics or are somewhat weaker should go to Gallian's book, which is also excellent. Students who are weaker still maybe should seek out Fraleigh.

Other reviewers are correct about the group theory being the strength of this book; ring and field theory are OK but short, but remember that this book is intended for a one semester undergraduate course. (Herstein was a ring theorist. It is natural to speculate that he chose the topics he did because of the course, not because of personal interest...) The optional topics (simplicity of A_n, Liouville's Criterion, etc.) are excellent.

"Topics in algebra" is supposed to be a year-long version of this book. That one is sometimes called "Herstein" and this one is "Baby Herstein". Happily though, Baby Herstein still has content, unlike "Baby Hungerford"...


Excellent Introduction to Abstract Algebra   December 5, 2004
Pioneer
8 out of 9 found this review helpful

My first introduction to abstract algebra has been by this book and I've found it to be an excellent choice. It is a concise book with lots of content. Topics are discussed very fluidly. One really gets the essence of the topics with a lot of insight. The book is simply too elegant. Another great asset of the book is its high quality exercises. Most of the exercises are difficult, nontrivial and provide further insight. This is the only abstract algebra book I've seen, but I don't think any other book could surpass this one in the quality of treatment.


baby Herstein!   May 1, 2004
Fourier Jr (Victoria, Canada)
7 out of 8 found this review helpful

I had this text for an intermediate course (after the 1st one) on abstract algebra including groups, rings, fields and homomorphisms, quotient structures, etc right up to where Galois Theory would start, and it was good for that. I wouldn't say that this book is good for someone who has never seen algebra before because the easy problems are still kind of hard compared with other books. If you've seen a bit of algebra before though this book would be really good. It's got tons of problems at the end of almost every section also.


Great Text for an Undergraduate   April 14, 1999
Christopher Long (San Diego, CA USA)
3 out of 3 found this review helpful

This was the book that really introduced me to how much fun mathematics could be. The main text covers all of the standard topics in a very clear manner, but the overwhelming strength of this text is the large number of excellent exercises, which range in difficulty from easy to graduate level. Most are labeled as to difficultly, which is unusual, but nice.

I cannot emphasize this enough: to get the most of this text you should do as many exercises as possible of a difficulty level that challenges you. If you do, you'll have a solid foundation of abstract algebra that will be more than sufficient if you choose to pursue graduate studies in math (or another field) later.