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I'm studying Herstein's Topics in algebra and Hardy&Wright's An introduction to the theory of numbers, and I was wondering if there are some exercise books (that is, books with solved problems and exercises) that I can use as companions.

The books I am searching for should be:

  • full of hard, non-obvious, non-common, and thought-provoking problems;
  • rich of complete, step by step, rigorous, and enlightening solutions.

Roughly speaking, something similar (in some sense) to George Pólya and Gabor Szegő's Problems and Theorems in Analysis.

Update: I would appreciate some suggestions in reference to the answers given to these questions too: 1 & 2

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    $\begingroup$ Related: 1, 2. $\endgroup$ – Hakim Sep 28 '14 at 9:51
  • $\begingroup$ @Hakim Thanks for your comment, but I'm lookinf for more suggestions. Do you have any other piece of advice? $\endgroup$ – Dal Sep 30 '14 at 19:37
  • $\begingroup$ Like what? The answers in the two questions I referenced are already sufficient. $\endgroup$ – Hakim Sep 30 '14 at 21:18
  • $\begingroup$ @Hakim Thinking better about it, I admit you are right. Thank you again very much. $\endgroup$ – Dal Sep 30 '14 at 21:23
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    $\begingroup$ You're welcome, good luck with your studies! :-) $\endgroup$ – Hakim Sep 30 '14 at 21:23
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You could try 'The Red Book of Mathematical Problems' and 'The Green Book of Mathematical Problems' by Kenneth Hardy and Kenneth S. Williams.

These contain numerous problems, with hints and solutions.

Available directly from Dover Publications.

An on-line selection of problems by Yang Wang is here, but no hints or solutions.

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There is a booklet floating around the internet titled " Problems in Elementary Number Theory".

Be warned, the problems in that booklet are VERY hard. Though it should supplement Hardy's book well if you are able to work through the problems.

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Abstract Algebra - Dummit and Foote is a good book to study and solve abstract algebra.

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