Can you please help with the following problem?
Let $K$ be a field and let $f(x) \in K[x]$ be a separable polynomial. If $E/K$ is a splitting field of $f(x)$, prove that every root of $f(x)$ in $E$ is a separable element over $K$.
Let $z$ be a root of $f(x)$ in $E$ and suppose $z$ is not a separable element over $K$, then by definition the minimal polynomial, say $h(x)$, of $z$ over $K$ is not separable, so we can find a repeated root $q \in E$ of $h(x)$. Now since $f(z)=0$ and $h(x)$ is the minimal polynomial of $z$ over $K$ then $h(x)|f(x)$ so that $q$ is also a repeated root of $f(x)$. This implies then that $f(x)$ is not separable in $E[x]$, this contradicts the fact that $f(x)$ is separable over K[x]$.
For this problem I'm using the fact that if $E/K$ is a field extension and $f(x) \in K[x]$ is a separable polynomial then $f(x)$ is separable when considered as an element of $E[x]$.