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I am currently self-studying abstract algebra from Artin. In that background, I am looking for a problem book in a spirit somewhat similar to Problems in Mathematical Analysis by AMS so that I have a lot of problems to solve.

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I am trying to collect problems with solutions on - there are also some problems from Artin. The database is still small but perhaps you can add the problems you're going to solve while studying abstract algebra. – Jernej Jul 24 '12 at 10:29
This question might be of interest:… – Francis Adams Jul 24 '12 at 21:10
Fraleigh's book is quite nice for self study because it (a) has lots and lots of problems, and (b) has complete solutions available. – user641 Jul 24 '12 at 23:49
Thanks Francis. The thread was useful. Fraleigh's book seems interesting. The solutions are exactly what I want. Since I am self-studying in isolation, it is useful to verify the solutions. – Jayesh Badwaik Jul 25 '12 at 7:24
up vote 13 down vote accepted

I don't know if I would call it a "problem book" but Herstein's fantastic "Topics in Algebra" has some amazing basic problems in abstract algebra. If you want more commutative algebra, Atiyah-Madonald have a lot too, but you should probably wait with that one.

For a problem book, I would recommend: Exercises in Algebra: A Collection of Exercises, in Algebra, Linear Algebra and Geometry (Algebra, Logic and Applications , Vol 6) .

Look for old quals in Algebra too and old exams. I can not, if you're learning abstract algebra, recommend Aluffi's book "Algebra: Chapter 0" enough. I used it as a first introduction and I fell in love with the subject. Good luck!

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+1 for Herstein. Certainly the way to go. – Shahab Jul 24 '12 at 11:58
Hi, thanks for your comments. Herstein does have a fresh perspective of problems and exposition. I read it and already like how he presented the three different proofs of Sylow Theorems including the very basic combinatorial one. Thanks – Jayesh Badwaik Jul 25 '12 at 7:20
+1 for Herstien. Still probably the single best collection of algebra exercises there is in one book. – Mathemagician1234 Dec 29 '12 at 6:52

While I think that doing all the problems in Artin is more than enough, you might want to look through the problems on algebra in the Berkeley Problems in Mathematics and see whether you can solve them reasonably quickly. The Berkeley Problems is not a book to look for fascinating algebra problems, though. Unfortunately, I don't know of any problem books in algebra. I think Artin actually has very good, not-so-standard problems for undergraduate algebra.

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I admit they are really good. Still, I believe it is better to look around just to see what problems do other people see around. I am not a math major and my college/university does not have these courses, so I am really scarce on exposure. – Jayesh Badwaik Jul 25 '12 at 7:22
I am now reading the Berkeley Problems in Mathematics and it seems good. Lets see how well can I solve the problems now. – Jayesh Badwaik Jul 25 '12 at 7:31
I unaccepted your answer and instead accepted Dedalus's answer since on retrospective, I have found some of the book Exercise in Algebra and other books a more concentrated source of problems. I hope you don't mind. – Jayesh Badwaik Oct 29 '12 at 18:23

For a first pass through the material, I really enjoyed Pinter's A Book of Abstract Algebra, as it presented the material quite well (although it did not go very deep). Also, I started out hating Algebra: Pure and Applied by Papantonopoulou, but I actually quite enjoyed it by the end of the course I used it for. Tons of problems in that one.

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I taught algebra from Pinter's book several times and really appreciated the problem sets. They cover a surprising amount of material in what could be described as bite-sized chunks. – Chris Leary Jul 24 '12 at 23:43

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