Let \(L/k\) be a field extension. An element \(\alpha\in L\) that is algebraic over \(k\) is separable over \(k\) if its minimal polynomial over \(k\) is separable. An algebraic extension \(L/k\) is separable if every \(\alpha\in L\) is separable over \(k\).
An algebraic extension \(L/k\) is separable if and only if every finite subextension \(k\subseteq L'\subseteq L\) is separable over \(k\).
If \(L/k\) is separable, every element of a subextension \(L'\) is an element of \(L\), hence separable over \(k\). Conversely, each \(\alpha\in L\) lies in the finite subextension \(k(\alpha)\), which is separable by hypothesis, so \(\alpha\) is separable over \(k\).