$E/F, K/F$ fields with $E\cap K = F$, $E/F$ Galois, $\alpha \in K$ with min poly $f(x) \in F[x]$ then $f$ irreducible over $E$.

Question

Let $E/F$ and $K/F$ be field extensions contained in some common field $L$. Suppose $E/F$ is Galois and that $E \cap K = F$. Let $\alpha \in K$ be algebraic over $F$ with minimal polynomial $f$. Show that $f$ is irreducible over $E$.

So far: I know that $f$ has no roots in $E$, since otherwise it would have all roots in $E$ by normality, whence $\alpha \in E$ (contradicting $E \cap K = F$). But I don't see why $f$ mustn't split in $E$ into irreducible factors.