The field $\mathbb{Q}$ satisfies the very nice property that if $F$ is a finite degree field extension of $\mathbb{Q}$, then $F / \mathbb{Q}$ is a separable field extension. That is to say, it is algebraic, and for every $\alpha \in F$, the minimal polynomial $m_{\alpha}(x)$ of $\alpha$ over $\mathbb{Q}$ has $\operatorname{deg}(m_{\alpha})$ distinct roots in a splitting field $L$ of $m_{\alpha}$ over $\mathbb{Q}$. This property can be extended to the theorem:
$\textbf{Theorem: }$ Any finite degree field extension of a field of characteristic zero is separable.
My question is does this property extend to any larger class of fields than just those of characteristic zero?