Rating: 
Summary: Best at what it is
Review: (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"...
Rating:
Summary: good book for 1st semester course
Review: Abstract algebra (AKA "algebraic structures", "modern algebra", or simply "algebra") can be a difficult topic depending on its presentation. The difficulty comes in the abstractness of the topic (generalizations that give us useful properties), not the complexity of the area (though, further study can provide some of this). 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...
Rating:
Summary: good book for 1st semester course
Review: Abstract algebra (AKA "algebraic structures", "modern algebra", or simply "algebra") can be a difficult topic depending on its presentation. The difficulty comes in the abstractness of the topic (generalizations that give us useful properties), not the complexity of the area (though, further study can provide some of this). 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...
Rating:
Summary: baby Herstein!
Review: 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.
Rating: 
Summary: Not complete enough
Review: I realize that Herstein dominated abstract algebra instruction for undergraduates for thirty years, but I don't find this text complete enough. I much prefer Gallian's text, which has begun to overtake undergraduate academia.
Rating: 
Summary: Not a bad book but I am sure it could be better.
Review: I want you first to know that I have only read about 3/4 of the book and I have stopped after field extentions. I am trying here to comment on the book from a relatively more advanced point of view because I have had all the subjects in depth in some other classes. I think Hersteins treatment of groups is more than excellent I would not recommend any other book for group theory at the undergraduate level. But he starts loosing this track in his treatment of rings, and I feel he starts getting faster and faster in explaing ideals and I do not think he did it very well. Field extension and Galois theory go even faster. I think you should stop reading the book after group theory and try some other book in the subject of ring theory something like Jacobson's "Basic Algebra I" for advanced students. But the book is not that bad if you can absorb things fast enough. It even has a chapter about straight edge and compass constructions which is a remarkable subject for me. It even has an optional chapter about the simplicity of the permutation group and some more results on finite abelian groups (If I am not mistaking).
Rating:
Summary: Excellent Introduction to Abstract Algebra
Review: 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.
Rating:
Summary: Great introduction to Abstract Algebra
Review: 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.
Rating:
Summary: Great Text for an Undergraduate
Review: 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.
Rating:
Summary: A great book
Review: We are so lucky that Herstein, as a great write, wrote this text in the last two years of his life. I think I am careful enough to say this is the best text I ever seen for years. Certainly, "best text" to different people are different. For every student who wants to be a mathematician, I stronly recommend this as the first text on abstract algebra. If you have a month, reading and doing all the problems, you will love mathematics. And I want to point out a mistake by another reviewer here. He said something about 30 years. However, this book is written 12 years ago.
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