I want to show that if $K$ is a field with characteristic $\operatorname{char} K = p > 0$ and $K \leq E$ is a field extension such that every intermediate field is Galois-closed then $K \leq E$ is algebraic and Galois.
A field $K \leq E$ is Galois-closed if and only if $K'' = K$ where $K' = \lbrace \sigma \in \operatorname{Aut}_K(E) \mid \forall \alpha \in K : \sigma(\alpha)= \alpha \rbrace$ and $K'' = \lbrace \alpha \in E \mid \forall \sigma \in K' : \sigma(\alpha) = \alpha \rbrace$.
An extension $K \leq E$ is called Galois if and only if for all $\alpha \in E \setminus K$ there exists $\sigma \in \operatorname{Aut}_K(E)$ such that $\sigma(\alpha) \neq \alpha$.
So far I proved that $K \leq E$ is separable and $u \in E$ with $u^{mp^k} \in K$ with $p \nmid m$ and $k \geq 1$ then $u^m \in K$. I tried using this to proof the statement by using transcendent elements $\alpha$ over $K$ and look at $K(\alpha^p)$ but I didn’t succeed.
Edit: I figured out that if $E/K$ is a finite field extension then an intermediate field $Z$ is Galois-closed iff $[E:Z] = |Gal(E/Z)|$. Also under the condition that $E/K$ is a finite field extension $E/K$ being Galois is equivalent to: every intermediate field of $E/K$ is Galois-closed. For the infinite case I was only able to show that if $E/K$ is algebraic then an intermediate field $Z$ of $E/K$ is Galois-closed iff $Z \subseteq E$ is Galois. I am not sure how to show that $E/K$ is algebraic in the infinite case. For the finite case $E/K$ is algebraic and I was able to show the result.