This is a follow-up to this question, where I am attempting to solve problem 5.26 of Morandis Field and Galois Theory.
Let $K/F$ be a normal extension, and let $E \supseteq K$ be some larger field, with $a \in E - K$ separable (and algebraic) over $F$. First of all, is it true that $K \cap F(a) = F$? And if so, is it then true that $[K(a):F(a)] = [K:F]$? My intuition tells me that this is too good to be true, but I can't find a counter-example.
Something that might help is that none of the roots of $\min(F,a)$ are in $K$. If one of them is, then $\min(F,a)$ splits over $K$, implying that $a \in K$, contrary to our assumption.
It is not true that $K\cap F(a)=F$. For a counterexample, take an element $b\in K-F$ that is not a square in $K$ and let $a\in E$ be such that $a^2=b$. Then $$F\subsetneq F(b)\subset K\cap F(a).$$