Condition for Intermediate Field to be Galois

Question

I was doing an exercise (25.3.8) in Shahriar Shahriari's book, Algebra in Action. The exercise states that $K$ be intermediate field $F \subset K \subset E$ with $E/F$ is a Galois extension. Prove that $K$ is Galois if and only if $\sigma(K)=K$ for all $\sigma \in \text{Gal}(E/F)$. I can prove if $ K $ Galois then $\sigma(K)=K$ for all $\sigma \in \text{Gal}(E/F)$ but I cannot prove the other direction. Any hint will be appreciated.

Answer 1

Let $p(x)$ in $F[x]$ be irreducible over $F$, and let $p(\alpha)=0$ for some $\alpha$ in $K$. Then $\alpha$ is in $E$, and $E$ is normal over $F$, so every conjugate of $\alpha$ over $F$ is in $E$, and for each such conjugate $\beta$ there exists $\sigma$ in the group of $E/F$ such that $\sigma(\alpha)=\beta$. Then $\sigma(K)=K$ implies $\beta$ is in $K$ for every conjugate $\beta$ of $\alpha$ over $F$, so $K$ is normal over $F$.