Every finite group is isomorphic to some Galois group for some finite normal extension of some field.
I'm trying to write a proof, but I know this is incorrect. Can anyone point me in the write direction?
Proposition: Every finite group is isomorphic to some Galois Group $\text{Aut}_F(K)$ for some finite normal extension K of some field F.
My hack at a proof: For every $n \in \mathbb{Z}^+$ where $n = |\text{Aut}_E K|$ there exists a normal extension $K$ of $E$ such that $\text{Aut}_E K \cong S_n$. Then consider a subgroup $H \leq \text{Aut}_E K$ where $H \cong \text{Aut}_E K$. It follows that $H$ is the Galois group of $K$ over $K_H$.