Requesting abstract algebra book recommendations [closed]

Solution 1:

I thought I would offer my opinion now that I have some experience. I am using Artin; it is excellent. The principles are clearly presented, and, what I find most beneficial, the discussion is designed to give an intuitive understanding as well. Concepts and proofs are clearly presented so you don't have to untangle them. When learning a new subject, I find it valuable to have a good grasp of the material and its ramifications. It leads to a sense of confidence and fulfillment.

The problems are divided into two levels of difficulty. Each section (~4 pages) has a small set of problems that are quite doable and enhance your understanding. Then each chapter ends with a set of more challenging problems.

The format (which I personally consider to be very important) is most accessible. Pages are not crammed. Key points are given adequate space so you can visually absorb them. And subscripts are easily seen.

If you really want to have a great learning experience, you can use Artin along with a parallel, free course from Harvard featuring Benedict Gross, which includes excellent videos and class notes. Really outstanding!

Here is the link:

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

In conclusion, I would also like to offer my personal experience with Dummit & Foote - which is not so endearing. I found the verbiage unnecessarily pedantic. Key principles are embedded in a large number of pages so it's not easy to focus on the salient features. Although it is ~10^3 pages long, some points that are elegantly proven in Artin are just left hanging (not even "left to the reader" or h.w. problems). The format is large pages crammed with smallish print with examples in tiny print. Granted, D&F is encyclopaedic in nature, but aside from furniture, it was the heaviest item I owned. May be good as a reference if you already know what's up.

$EDIT$ : Some time and further along, I would substantially modify my opinions of both "Artin" and "D & F."

Artin is still much better to learn group theory with. But after that, Dummit takes over the exposition in D & F and the presentation really takes flight. On the other hand, Artin no longer provides the intuitive insight or as extensive a presentation that Dummit provides in very accessible form.

Solution 2:

I think A first course in abstract algebra by Fraleigh is a good textbook for self study and there is also a solution manual.

Solution 3:

I see that many people have recommended the Artin text for self study. I used the second edition of this text for an algebra course I did at university and I would definitely NOT recommend this for self study. Let me explain why:

(1) The exercises at the end of each chapter are very often not related to the material. They are either routine or do not expand further on your understanding (I encourage you, if possible to see what I mean by looking at chapter 2 of the 2nd edition on exercises about the quotient group).

(2) If you look at chapter 10 on Linear Representations of Finite Groups, Artin presents a proof on orthogonality of characters which uses a Lemma proved using continuity that is not entirely rigorous (in fact I have asked a question on this site regarding the lemma). Even though Serre's book of the same name as this chapter is a graduate text I found the proof there to be way easier to understand (the proof there on the orthogonality of characters is entirely rigorous).

(3) In fact my lecturer for the course said that in future if he ever uses the book again, he will ditch the continuity bit of chapter 5. It is not rigorous and I do not encourage you to learn the proof of things like the Cayley - Hamilton Theorem the way Artin has done it (please see Axler's Linear Algebra Done Right, chapter 8 for a more rigorous proof.

(4) If you want to learn Ring Theory, do not read chapter 11 of Artin's book for you will be very confused - in particular the sections on adjoining elements to a ring and product rings. I clearly remember spending at least 2 hours on an example that Artin gave of a ring adjoined with some element (I don't remember of the top of my head what the original ring was, but I remember it being isomorphic to $\mathbb{F}_5$). I had to use a combination of the first isomorphism and lattice theorems to understand what he was saying - I don't think many people in my class got what he was saying too.

I frequently had to refer to Herstein's Abstract Algebra (not Topics in Algebra) and in fact that was what I used most in the end. Let me explain why this book is much better than Artin's Algebra and more useful for self study.

(1) Herstein gives many examples - the construction of the quotient group is famous for being difficult to understand. Herstein explains this section beautifully. To top it off, the exercises at the end are even better for they expand on the concept of the quotient ring even more. For example, Herstein asks what is $\mathbb{R}/\mathbb{Z}$ isomorphic to? Or for that matter what is $\mathbb{R}^2$ mod out all the lattice points isomorphic to? The point is that the exercises at the end $will$ strengthen your understanding and showing the many connections that the quotient group has to other things.

(2) On the sections on Sylow Theory, Herstein points out very clearly the technique of induction that he will use over and over again. I found this very encouraging and helpful from him to share experience like that. Often many authors (like Artin) don't tell you the techniques of finite group theory that are important.

I could go on typing, but I hope that all of the above helps!!