Let $K$ be a fixed number field, $d>0$ a fixed natural number and $S\subset \text{Spec}(O_K)$ a fixed finite subset of primes.
I'm trying to prove the following statement:
There are finitely many possibilies for a number field $K\subset F\subset \overline{K}$ such that $[F:K]\leq d$ and such that $F/K$ is unramified for all $\mathfrak{p}\notin S$.
Here's what I know: there is a theorem by Hermite which says that there are finitely many number fields with a fixed discriminant. Besides, I know the discriminant of an extension says something about ramification.
It looks like a good start, but I still can't figure out how to make it work. I don't see how to relate the condition $[F:K]\leq d$ with the discriminant and I can't see how am I going to apply the finiteness of $S$.
The key here is that the exponents of the ramified primes appearing in the factorization of the discriminants are bounded by a function of $ d $ (Neukirch, Theorems 2.6 and 2.9 in chapter III), so the condition that $ F/K $ is unramified outside $ S $, which is a finite set, tells you that there are only finitely many possibilities for the discriminant of $ F $. This finishes the proof along with Hermite's theorem.