TITLE: Learning Algebra & Group Theory on my own
QUESTION [12 upvotes]: I'm learning Algebra & Group Theory, casually, on my own.  Professionally, I'm a computer consultant, with a growing interest in the mathematical and theoretical aspects.  I've been amazed with the applications of Algebra to CS things like cryptography, coding theory, and combinatorial search.
I'm using Artin's book, and, as high quality as it is, I find it leaves me dry.  A lot of names for new concepts, and proving that one thing follows from others.  But I don't see the powerful tool I envisioned that I'd gain taking shape in my hand.
Perhaps because I'm learning independently and casually, I need a book which will give me tools, showing me how they can be used to solve concrete problems outside of algebra.
What book(s) do you recommend?

REPLY [2 votes]: First, my background - a math major and a retired software engineer.
It takes time and efforts to go from theory to application. You need to convert yourself from a computer scientist to a mathematician. Then convert back to computer scientist. A hard process.  Please be patient and give yourself more time.
All the books the answerers recommend seem to be good although I did not read all of them. I am reading Dummit and Foote’s Abstract Algebra because I am studying for math, not for computers. Please note the word I use – ‘for’.
To answer your particular question, I would recommend P.M. Cohn’s two volumes – Basic Algebra and Further Algebra and Applications.  These two books contain enough theory and also pay attention to applications. Cohn tried to tell the reader how to apply the theory. You may not find too much number theory in those two books. For that, you need a good number theory book.
EDIT:
If I were you, I would go to a university library and take a glance at all those books people here recommend before I buy any book. Everybody has his own taste.