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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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.
Rudolf Lidl, Gunter Pilz "Applied Abstract Algebra" (Undergraduate Texts in Mathematics) [From Contents: Applications of Boolean Algebras, Coding Theory, Cryptology, Image Understanding]
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