I am studying the splitting field for $x^4-2\in\mathbb{Q}[x]$. Let $F$ denote the splitting field for $x^4-2\in\mathbb{Q}[x]$. I found that the Galois group of the extension $\mathbb{Q}\subset F$ is $D_8.$ And since there are four proper normal subgroups of $D_8$ we can conclude that there are four subfields of $F.$ Four subgroups are following: trivial subgroup, $\mathbb{Z}_2$, Klein four group and $\mathbb{Z}_4$.
Now I want to know which of these subfields are stable.
Definition of stable: if $E$ is an intermediate field of the extension $K\subset F$, $E$ is said to be stable (relative to $K$ and $F$) if every $K$-automorphism $\sigma\in Aut_KF$ maps $E$ into itself.
Then by the definition of stable, trivial subgroup and $\mathbb{Z}_2$ are stable? Am I right?