How can I show that every finite group is the Galois group of an extension $K/F$ where $F$ is itself a finite extension of $\mathbb Q$?
I know the following:
Every finite group is contained in $S_p$ for a large enough prime $p$.
Every irreducible polynomial in $\mathbb Q[x]$ of degree $p$ having exactly $p-2$ real roots has a Galois group $S_p$ over $\mathbb Q$.
For any $n$ there is an irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots.
Does this have something to do with the inverse Galois problem?