You have seen an introduction to Galois theory; of course, there is more. A deeper study of abelian fields, that is, fields having (possibly infinite) abelian Galois groups, begins with Kummer theory and continues on to class field theory. Infinite Galois groups are topologized, and there is a bijection between intermediate fields and closed subgroups. The theorems are of basic importance in algebraic number theory. There is also a Galois theory classifying division algebras (see [Jacobson (1956)] and a Galois theory classifying commutative rings (see [Chase, Harrison, Rosenberg]).