Let $F\subseteq E\subseteq M$ be finite field extensions. Assume if $M/F$ is normal. Prove that $M/E$ is normal.
Attampt:
I have proved that $M/E$ is algebric extension. Let $a\in M$. Let $f_T=\text{irr}(a,T)$ where $T=E,F$. If $f_F=f_E$ then we are done. Else, $f_F\ne f_E$. Not sure how to proceed.
There is a known theorem that a finite field extension $L/K$ is normal if and only if there is a polynomial in $K[x]$ such that $L$ is its splitting field. So in your case $M$ is the splitting field of a polynomial $f\in F[x]$. But since $F\subseteq E$ we get $f\in E[x]$. So $M$ is also a splitting field of a polynomial in $E[x]$. Hence $M/E$ is a normal extension.