Proposition. $L / F$ is Galois $\implies$ $L / K$ is Galois [22017]

    Let $K / F$ and $L / K$ be algebraic field extensions. Then $L / F$ is Galois $\implies$ $L / K$ is Galois.

    Proof. sketch [22017A]

      Suppose $L / F$ is Galois. Then by Proposition 17011, $L / K$ is separable.

      We use (5) of Proposition 2205 to show that $L / K$ is normal. Fix an algebraic closure of $F$ such that $F \subseteq K \subseteq L \subseteq \overline{F}$. If $\varphi : L \to \overline{F}$ is a $K$-embeddings, then it is an $F$-embeddings. By Proposition 2205, $\varphi(L) = L$. Again by Proposition 2205, $\varphi(L) = L$ for all $K$-embedding $\varphi : L \to \overline{F}$ of $L$ $\implies$ $L / K$ is normal.