Algebra Comments about "Topics in Algebra" by I.N. Hertsein?

[bacte2013]

Dear Physics Forum advisers,

Today, I got two gifts from my research mentor: "Topics in Algebra" by I.N. Herstein and "Abstract Algebra" by Dummit/Foote. I am very happy and grateful for his gifts, but I already have been studying the abstract algebra through Michael Artin and Hoffman/Kunze. I went through Herstein and D/K, and it seems that both cover the more or less same topics covered by Artin, and certainly not as detailed as H/K in terms of the linear algebra. However, I like the exposition and details of Herstein and D/F. What is your opinion on those books? Should I keep focus on Artin and H/K?

 

[micromass]

The books you listed are not comparable. Dummit and Foote/ Herstein cover way more than Artin. Artin is meant to be a (very good) introduction to abstract algebra, but you cannot compare it to more advanced works like Herstein and Dummit and Foote. Those are books which are used in some graduate programs (but are typically meant for undergrad).
Hoffman and Kunze is a very good and solid linear algebra book. It is not an abstract algebra book and thus is not comparable to Dummitand Foote, Herstein or Artin.

 

[bacte2013]

Thanks for the comment. I have been studying Artin and Hoffman/Kunze. Do you personally recommend Herstein and D/F over Artin? Or am I doing the right track?

 

[micromass]

Depends on you. I cannot make that choice for you. I would personally use neither of those books since neither is my style. But your style might be very different from mine.

 

[bacte2013]

Just curious, are you familiar with "Algebra" by MacLane/Birkhoff and "A Survey of Modern Algebra" by Birkhoff/MacLane?

 

[micromass]

Yes.

 

[bacte2013]

How are they? Is it suitable to study them after Artin?

 

[bacte2013]

There are so many books I want to read, but only limited amount of time.

 

[micromass]

I really don't know how to answer this. It depends entirely on you and what you are trying to do. So for some people asking this question, the answer is no, for others the answer is yes. All I can say is that both Birkhoff and MacLane are truly top mathematicians, and that their book is very solid, contains very nice exercises and you cannot go wrong with them. Whether this book suits your purposes, whether you agree with the philosophy of the book, whether you will find it too easy or too hard, that are things I cannot answer for you without knowing you a lot better.

 

[bacte2013]

Dear Professor micromass, can I send a PM to you to discuss about the problem more in-depth?

 

[micromass]

Sure. (and although I teach at university, I do not have the title of professor yet).

 

[mathwonk]

another opinion:

These books are all very different. I like Artin, which I consider to cover more than Herstein, although there is no containment relation between them in either direction. I also feel Artin's explanations are more insightful than Herstein's, but that is perhaps a personal matter. Dummitt and Foote covers more than either of the other books, although again there are topics in both Artin and Herstein that are not covered in DF. DF can be used for probably two courses, one undergrad and then also a grad sequel. I myself would also use Artin in a beginning grad course but not Herstein, which I consider an undergrad book.

Birkhoff and Maclane is very old fashioned but useful for that very reason for concreteness, while Maclane and Birkhoff is much more advanced and abstract. I have taught from most of these books at university level. In my opinion Herstein is the least useful of all these titles to prepare for grad work , but the problem sets are fun, and the treatment of normal forms for some linear transformations is more complete than many. The last 25 pages also treat three advanced topics that are not found everywhere.

But I suggest you should use the book that speaks most clearly to you, and maybe dip into others that seem more difficult as your experience grows. Many people consider DF the standard text today, but i think you will miss some things if that is all you know, that would be made clear in Artin.

 

[mpresic]

Herstein's text Topics in Algebra has a great reputation among mathematicians. The possible defect with the book is the problems. Herstein notes in his preface is the problems are presented, not so much to solve, but to be tackled. The trouble is you are never sure you should be able to solve any particular problem or it is too advanced. For example, one problem in Chapter 2 after Homomorphisms are presented says show any group of order 9 is abelian. Later in the text, Herstein presents a proof that any group of order p-squared is abelian; p is a prime number (like 3). The trouble is it is not obvious you can solve the first problem with the techniques presented in the text with the methods presented up to that point. Maybe you can't because this was one meant to be tackled rather than solved. It is difficult to assess whether you know the material from your performance on some of the problems.

On the other hand where else can you find 3 proofs of Sylow's theorem. I did find Artin was easier. (I think practically any treatment even very advanced works would be easier than Herstein's problems)