What is separable polynomial?

A polynomial $P \in K[X]$ is separable if it has no repeated root in a splitting field — equivalently $\gcd(P, P') = 1$ (a repeated root is a common root; conversely, over the splitting field, a common root is repeated; and the gcd does not change under field extension, Corollary 3.17’s argument). An algebraic element is separable if its minimal polynomial is; an extension $L/K$ is separable if all its elements are.