Since every finite group $G$ is embedded in $S_n$ for $n = |G|$ and Hilbert showed that $S_n$ appears as a Galois group of $K/\Bbb{Q}$ for some Galois extension $K$, then how does that not wrap up the Inverse Galois Problem?
Why couldn't we somehow take any subgroup of $S_n$ and show that it must be the Galois group over $\Bbb{Q}$ of some $L \supset K$?
Best Answer
Subgroups $H$ of $Gal(E/F)$ correspond to intermediate fields $K$ where $H = Gal(E/K)$ Therefore, if we apply it to $F = \mathbb{Q}$, then all we can conclude is that $H$ is the Galois group of some field extension $E/K$, neither of which are required to be $\mathbb{Q}$.
To solve the inverse Galois problem, it is sufficient to show that every finite group is a quotient of $S_n$, not a subgroup.