I don't know about number theory, but Dummit & Foote, Hungerford, and Lang treat Galois theory and commutative algebra at the beginning/intermediate graduate level.
I was wondering if anyone can point me in the direction of any good texts on the following topics:
Galois Theory : I've done a course on this already but it was fairly elementary, so a slightly more advanced text might be good.
Algebraic Number Theory: Again I have studied the basics of this (mainly from a book by Stewart and Tall - "Algebraic Number Theory and Fermat's Last Theorem").
Commutative Algebra: I've studied the basics of ring/module theory but not too sure what comes next.
If anyone can recommend any other similar interesting areas of advanced undergrad/post grad abstract algebra (I'm sure there are loads) that I could have a look at I'd really appreciate it.
Thanks in advance,
For Galois Theory I recommend: A First Course In Abstract Algebra by John B. Fraleigh.
Amazon.com: A First Course in Abstract Algebra (6th Edition) (9780201335965): John B. Fraleigh: Books
For supplement you could use: Emil Artin's Notes on Galois Theory
Amazon.com: Galois Theory: Lectures Delivered at the University of Notre Dame by Emil Artin (Notre Dame Mathematical Lectures, Number 2) (9780486623429): Emil Artin, Arthur N. Milgram: Books
Or the text I used: A course in Galois Theory: D.J.H Garling
Amazon.com: A Course in Galois Theory (9780521312493): D. J. H. Garling: Books
but those two are very terse and omit a lot of details in proofs.
For Algebraic Number Theory: I used Introductory Algebraic Number Theory by Alaca and Williams
Amazon.com: Introductory Algebraic Number Theory (9780521540117): Saban Alaca, Kenneth S. Williams: Books
I love this book!!!
Finally for Commutative Algebra: Algebra: A graduate course by Issacs
Amazon.com: Algebra: A Graduate Course (Mathematics) (9780534190026): I. Martin Isaacs: Books
My take would be
Dummit and Foote-Abstract Algebra, Pierre Grillet-Abstract Algebra, or Lang-Algebra for Galois theory (Artin's book is ok, but as the previous poster said, very very short (I think less than 80 pages)).
For algebraic number theory I'd used either Lang's book or assuming you have the background you should use Milne's Notes (his other notes are also easily found and supplement the alg. number theory ones).
For commutative algebra I'd always say look at Atiyah and Macdonald first and then Eisenbud second (if neither of those tickle you you could look at Matsumoro). If you are looking for a more rounded (less specific) approach to commutative algebra I'd look at Dummit and Foote again.