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I've taken up self-study of math. (How smart can that be?) I've just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus.

I was thinking of trying abstract algebra. I would appreciate any book recommendations.

Thanks in advance.

Andrew

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this should be community wiki –  user9413 Jul 3 '11 at 18:15
    
I've converted the question to community wiki, as it's asking for a big list of things and there's no single right answer. –  Zev Chonoles Jul 3 '11 at 18:21
    
A comment of t.b. to this related question links to $3$ other related questions. –  Pierre-Yves Gaillard Jan 4 '12 at 18:23
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Just to spread this around a bit: Linear Algebra by Hoffman and Kunze –  Peter Sheldrick Aug 12 '12 at 23:47
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11 Answers

up vote 15 down vote accepted

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.

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+1 especially for the link above to the (open) course. –  Giorgio Mossa Jan 4 '12 at 18:33
    
+1 I have watched Gross's lectures 3 times for the sheer fun of it. Great stuff! –  Byron Schmuland Jan 4 '12 at 20:39
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I think A first course in abstract algebra by Fraleigh is a good textbook for self study and there is also a solution manual.

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Well, there are many. These are the ones which I have seen.

  • A first course in Abstract Algebra by John. B Fraleigh.

  • Contemporary abstract Algebra by Joseph Gallian.

  • Abstract Algebra by Dummit and Foote.

  • Algebra by Michael Artin.

  • Lectures in Abstract Algebra by Nathan Jacobson.

  • Topics in Algebra by I.N.Herstein.

For more you can see this link:

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Thanks to all. Sorry to take so long to express my appreciation to you very helpful suggestions. –  Andrew Sep 15 '11 at 21:50
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I wish you'd given some information about the real analysis book(s) you read to give us some idea of the level of your mathematical maturity. But without that, and based on the assumption that you were reasonably comfortable doing non-routine problems in your real analysis study, I'd say Fraleigh and Gallian would be good places to start.

They're both at the same level, and cover similar topics. Fraleigh breaks more difficult problems into smaller pieces which is perhaps better for self-study, but based on what I remember from a long time ago (and therefore not very reliable!), his writing style is a tiny bit dry. On the other hand, Gallian brings history and applications into the mix, which makes it more interesting. I remember that the first time I learned about the algebraic features of barcodes was in Gallian's book.

You might be able to find inexpensive used copies of the second to last editions of both, in which case I'd recommend you to get them both. In my experience having more than one book in a subject is always useful. Sometimes a proof or discussion is done much better in one than the other one.

I've seen all but Artin's among the other books that Chandru mentions, but I think they're more appropriate for a second course than the first.

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I think of Fraleigh as the opposite of dry. I found his book to have a friendly, conversational tone. –  MJD Aug 13 '12 at 2:14
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The book "Abstract Algebra: A Concrete Introduction" by Redfield develops abstract algebra with the purpose of showing the quintic is not solvable. It's like a math book with a plot. The last part is dedicated to the classification of all finite groups of order less than or equal to sixteen.

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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!!

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If you were good enough for an algebra class that used Artin,Ben-you should strongly consider working through Herstien's TOPICS IN ALGEBRA. It's the book I learned algebra from and it's still a classic with unmatched problem sets for the honor student to chew on. Give it a try. –  Mathemagician1234 Jan 6 '12 at 5:09
    
@Mathemagician1234 Why, is Artin's text too hard? I am learning some commutative ring theory now. By the way do you know where is good to start learning about modules? I find Paul Cohn's Introduction to Ring Theory too abstract, because he does it in a way that is very category theory flavoured. Could you recommend where to learn modules besides Isaacs' Algebra and Dummit and Foote? –  fpqc Jan 6 '12 at 12:43
    
Well,first,it's not that Artin is too hard- Herstien's TOPICS is about the same level and if anything,the exercises are harder.It's just that Artin doesn't really organize his book well and some sections aren't very rigorous. This makes the book hard to comprehend for a beginner. 2 great books for learning module theory are T.S. Blyth's MODULE THEORY: AN APPROACH TO LINEAR ALGEBRA and Hawkes and Hartley's RINGS,MODULES AND LINEAR ALGEBRA. The latter is a bit easier but less comprehensive-it doesn't cover much,but what it does cover,it covers beautifully.(continued) –  Mathemagician1234 Jan 6 '12 at 21:43
    
(continued) Blyth's book is comprehensive and very rigorous-it's a must have for anyone serious about algebra. For commutative ring theory,by the way,I reviewed a wonderful book for undergraduates for the MAA last year:TOPICS IN COMMUTATIVE RING THEORY by John J.Watkins. Beautifully and gently written with many examples and fascinating historical notes,it's an absolute gem for any beginner in serious mathematics.I STRONGLY recommend you find a copy-it will make your study of this great branch of algebra SO much easier,trust me. –  Mathemagician1234 Jan 6 '12 at 21:49
    
@BenjaminLimObviously you are way ahead of me, so my enthusiasm for Artin is probably from a different perspective. But as a defender of the faith, I actually liked the few problems in Ch 2 on quotient groups. It was nice to see something besides the integers and modular arithmetic. Plus as a self-studier, as I mentioned above, studying from Artin is greatly enhanced by Prof. Gross's lectures which are really something. The last lectures (from around the high 20's) deal with ring theory presented by a real master and a fabulous lecturer as well. They may be of interest to you. Regards. –  Andrew Jan 8 '12 at 21:23
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A great online resource is a book by Robert Ash from the University of Illinois in Urbana-Champaign. One (very) useful thing about this is that it has solutions for all exercices, which makes it great for self-study.

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Judson, Abstract Algebra: Theory and Applications, has illustrations, plentiful examples, end-of-chapter problems, and discussions of applications. It's available both in print and for free online.

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We use Dummit and Foote's book in our class. What was helpful was that a lot of bloggers post solutions to exercise problems.

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Lots of other good recommendations here,but the best one I've seen for self study is missing. A Course In Algebra by E.B. Vinberg is my absolute favorite algebra textbook. Extremely gentle yet covering algebra to the threshold of graduate level, Vinberg's book is loaded with examples from geometry,physics and other areas of mathematics. It also has literally hundreds of exercises. It's similar to Artin in spirit by having a geometric bent, but it's much more readable and accessible to the beginner. If I could only choose one algebra book for self-study,THAT would be it.

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I see my fan club is at it again. –  Mathemagician1234 Jul 1 '12 at 5:05
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I just did a search for "free" and it seemed this has yet to be mentioned

I just completed a semester long course that covered a good chunk of the free, recently updated as of August 2011, abstract algebra book that can be found here.

Each chapter ends with around 30 problems, the first few are usually leading computational questions that hint at things to be proven later or showcase oddities that you might not immediately see. The back of the book offers solutions to some of the problems, including a few from the beginning so you can be sure you've done them correctly. This is a plus for self study as I've been going through the sections we didn't cover.

There is still at least one typo and our professor deviated from the book a few times for idiosyncratic/notational reasons.

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