Tower of separable field extensions is separable

Question

Let $K ⊆ L ⊆ M$ be a tower of algebraic field extensions. Suppose $L/K$ and $M/L$ are separable extensions. Let $m ∈ M$. Then the minimal polynomial $f^m_K = q · f^m_L = q \cdot \prod_{i=1}^n (X-\alpha_i)$ for some $q ∈ L[X]$ and all $\alpha_i ∈ L$ different. Unfortunately all theorems concern finite field extensions, not arbitrary algebraic ones. So my guess is, to show that $f^m_K$ is separable, we must reduce the problem somehow to a finite field extension. Perhaps adjoining the coefficients of $q$ to $K$?