If $E/K$ is normal and $K \subseteq F \subseteq E$ then $E/F$ is normal.

Question

Assume that $E/K$ is a normal extension. Given the tower $K \subseteq F \subseteq E$ proof that $E/F$ needs to be normal.

I would like to use the characterization of normal extensions $F/K$ that says that any homomorphism $\sigma:F \to \overline{K}$ that fixes $K$ gives $\sigma(F) = F$. Here $\overline{K}$ is the algebraic closure of $K$. I have some notes that do so but they do not convince me...

Can you figure out how to prove it using this characterization?

Answer 1

Let $\sigma: E \to \overline{F}$ be an embedding that fixes $F$. Note that $\overline{F} = \overline{K}, \sigma$ can be viewed as map $\sigma: E \to \overline{K}$. Since $K \subseteq F$, and $\sigma$ fixes $F$, it also in particular fixes $K$. Can you proceed from here?