$E \supset K \supset k$, where $K \supset k$ is not algebraic. If $\alpha \in E$ is separable over K, .......

Question

$E \supset K \supset k$, where $K \supset k$ is not algebraic. If $\alpha \in E$ is separable over K, show that $\alpha$ is separable over a finitely generated extension of $k$

Answer 1

Proof impression: Consider the coefficients of the minimal polynomial of $\alpha$ over $K$.