How to find subfields of a non-Galois field extension? Let $K/F$ be a finite field extension.
If $K/F$ is Galois then it is well known that there is a bijection
between subgroups of $Gal(K/F)$ and subfields of $K/F$.
Since finding subgroups of a finite group is always easy (at least
in the meaning that we can find every subgroup by brute-force or otherwise)
this gives a nice way of finding subfields and proving they are the
only ones.
What can we do in the case that $K/F$ is not a Galois extension ?
that is: How can I find all subfields of a non-Galois field extension
?

In the inseparable case there is an idea for a substitute Galois correspondence due, I think, to Jacobson: instead of considering subgroups of the Galois group, we consider (restricted) Lie subalgebras of the Lie algebra of derivations. I don't know much about this approach, but "inseparable Galois theory" seems to be a good search term. 
