TITLE: Galois theory in reverse
QUESTION [5 upvotes]: Let $K/F$ be a Galois extension. The fundamental theorem of Galois theory says that there is a bijection between subfields of $K$ containing $F$ and subgroups $H$ of $G$, where $G$ is the Galois group $\text{Gal}(K/F)$. This theorem lets you start with some arbitrary field, something you can just pretty much pull out of thin air, and learn something about it using group theory. However, the groups used in this group theory are not arbitrary. You have to restrict your attention to the Galois group, which depends on the field and some subfield.
My question is, how do you do this process in reverse? Start with some arbitrary group (in the sense that its existence doesn't depend on anything) and learn something about it using field theory?

REPLY [4 votes]: This problem you are intereted in is call the inverse Galois problem.
Every finite group is a Galois group, every profinite group is a Galois group. In general, it is quite hard, for a general group $G$, to say if $G$ is a Galois group or not. I would recommand JP Serre's "Topics in Galois theory" for an authorative reference and state of the art (a bit old, but it is good for a starter).