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One of my problems with Abstract Algebra was that it was well... abstract. I could infer from doing problems "Oh this is useful for doing math with objects I'm not typically used to working with". However, most of what I did felt irrelevant and almost boring. Is there a good book that provides abstract algebra with some context? For example, one that incorporates the problems that mathematicians were trying to solve when developing it?

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nonlinearism answers the question below with groups in mind. But what about ring theory? History of math always seems a little fuzzy to me, but I think much of ring theory was developed for algebraic number theory and more specifically, Fermat's last theorem. Is there a book that develops algebraic number theory alongside introducing abstract algebra or perhaps more specifically, ring theory? –  RghtHndSd Aug 13 '13 at 19:27
    
Abstract algebra is literally all over the place in "higher" math. Therefore, my suggestion would be to learn more of your favorite field or perhaps look at it with an algebraic eye, as opposed to looking for a book full of tidbits of applications. –  Karl Kronenfeld Aug 13 '13 at 20:35
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3 Answers

Yes, look at Pinter's "A book of abstract algrebra". The whole point of the book is to start with very basics and prove the unsolvability of the quintic equation.

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+1 for a very nice recommendation. I would strongly suggest that, if that book is chosen, most / all of the exercises should be done. Pinter develops a number of important results (Sylow theorems, classification of finite abelian groups, etc.) in chains of exercises that span several chapters. –  T. Bongers Aug 13 '13 at 19:15
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Rudolf Lidl, Gunter Pilz "Applied Abstract Algebra" (Undergraduate Texts in Mathematics) [From Contents: Applications of Boolean Algebras, Coding Theory, Cryptology, Image Understanding]

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A few references for you:

Ash, R. B. A Primer of Abstract Mathematics. Washington, DC: Math. Assoc. Amer., 1998.

Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998.

Fraleigh, J. B. A First Course in Abstract Algebra, 7th ed. Reading, MA: Addison-Wesley, 2002.

Also, if you are interested in online resources, there is a 'book' online "Abstract Algebra Online" - also provides more references and links.

Hope this helps

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