A finite extension is separable exactly when separable generators exist and the separable degree equals the degree.Suggest a slogan
Let \(L/k\) be a finite field extension, and let \([L:k]_s\) be its separable degree, the number of \(k\)-embeddings \(L \to \bar{k}\) into an algebraic closure \(\bar{k}\) of \(k\). The following are equivalent:
\(L/k\) is separable.
\(L = k(\alpha)\) for some \(\alpha \in L\) separable over \(k\).
\(L = k(S)\) for some set \(S \subseteq L\) of elements separable over \(k\).
\([L:k]_s = [L:k]\).