Let $F\subset E\subset K$. Let $\alpha\in K$ be algebraic both over $F$ and $E$.
I want to show that $\alpha\in K/F$ be separable means $\alpha\in K/E$ is separable.
As an intermediate result, I'm trying to show that if $p(x)\in F[x]$ and $q(x)\in E[x]$ are the minimal polynomials, then $q(x) \mid p(x)$.
Intuitively, I feel this should be true, since $F$ lies inside $E$ and therefore its minimal polynomial should be some reduced (or not) form of $p(x)$. However, I don't know how to go about this.
Edit: The minimal polynomial for $\alpha\in E/F$ is the unique monic irreducible polynomial $p\in F[x]$ with $\alpha$ as a root.