A good introduction to elementary algebra? Is there a good book (or online text) that teaches elementary algebra?
The text should focus primarily on practical computation rather than on abstract theory (while still present full proofs). For example, i want to compute the minimal polynomial of the algebraic number $\sqrt[\leftroot{-1}\uproot{1}3]{5}+\sqrt[\leftroot{-1}\uproot{1}5]{3}$. How do i do that?
Most algebra books i have seen are far too abstract. They dive deep into abstractions and lose touch with elementary algebraic questions. I don't want to first learn about categories, exact sequences and the like, just to be able to solve some supposedly simple concrete algebra exercises.
 A: I'm a fan of Artin's Algebra.  J. S. Milne also has some online course notes covering group theory, field theory, and Galois theory which are quite nice.
It's worth pointing out that, distasteful though they may be right now, many abstractions such as exact sequences are defined precisely in order to help us compute things.  One could even say that the purpose of algebra is to define abstractions that help with computations.  You have every right to demand motivation for such things when they appear, but you shouldn't avoid them entirely.
A: I think Gallian's book is a great intro to abstract algebra. He is incredibly thorough in his dealing with group theory. He includes several numeric computation exercises but it also has it fair share of classic proofs. So if you're looking for intro to abstract algebra then I would definitely recommend his book. Someone mentioned Artin and Milne. Artin is certainly a good book, but depending on your level and interest it might not be the best. As for Milne, well, to be fair I've never seen his Algebra notes, but I have seen 3 others (ANT, CFT, and Complex Multiplication) and they aren't exactly kind to the new reader. Finally, as for the computational number theory, I would like to second Shoup's book mentioned above
A: I think this is what you want ABSTRACT ALGEBRA ONLINE STUDY GUIDE.
A: Victor Shoup's a Computational Introduction to Abstract Algebra and Number Theory  may be to your liking. In particular chapter 18 covers computing the minimum polynomial.
