Let $f(x)$ be a quartic polynomial over $\mathbb{Q}$, and let $K$ be the splitting field of $f(x)$ over $\mathbb{Q}$.
I know that the following facts:
If $f(x)$ is separable over $\mathbb{Q}$, then the Galois group $G_{f}$ of $f(x)$ is isomorphic to the symmetric group $S_{4}$.
$f(x)$ is irreducible over $\mathbb{Q}$ if and only if $G_{f}$ is isomorphic to a transitive subgroup of the symmetric group $S_{4}$.
What can we say about that the Galois group for an arbitrary quartic polynomial over $\mathbb{Q}$?
Some problem claims that if $f(x)$ is a quartic polynomial over $\mathbb{Q}$ with splitting field $K$ satisfying $[K:\mathbb{Q}]=8$, then $G_{f}\cong D_{4}$, where $D_{4}$ is the dihedral group of order $8$.
I think we need the condition separability or irreducibility of $f(x)$ over $\mathbb{Q}$ is needed, but there's nothing like that.
Do I miss something or have a misconception? Give some advice or comments. Thank you!