# Advanced galois theory/field theory book suggestions

I have done the equivalent of an undergraduate algebra course(rings/fields/groups) and have read Stewart's Galois theory book. Galois theory was a lot of fun and I would like to continue studying it but I have no idea how to progress studying it or what the big theorems/questions further are.

I would like any suggestions on books that extend basic galois theory. All the suggestions I can find are at the level of Stewart's book(ie. end with essentially a proof of the insolubility of the quintic).

• There is some deeper Galois theory in Isaac's Algebra. His treatment culminates in the Artin-Schrier theorem, which says that when the algebraic closure of a field is a finite degree extension, then it is very similar to the extension $|C:R|$. Another neat topic Galois theoretic topic is Hilbert's ramification theory, which you can find treated in the first (extremely dense) chapter of Neukirch's Algebraic Number Theory. I would recommend reading the sections on Galois theory and commutative algebra in Isaac's before reading Neukirch, since some of the basic exposition in Neukirch is lacking. – Lorenzo Najt Jun 12 '14 at 0:04
• Galois theory is a major underpinning of algebraic number theory as a whole, so you might like to go on to see a broader slice of that subject. I recommend all of J.S. Milne's course notes for number theory and algebraic geometry from the level you're already at through topics like etale cohomology that are rarely lectured. – Kevin Carlson Jun 12 '14 at 0:08