Let $K$ be a field.
Let $\map P X \in K \sqbrk X$ be a polynomial of degree $n$.
$P$ is separable if and only if its roots are distinct in an algebraic closure of $K$.
$P$ is separable if and only if it has no double roots in every field extension of $K$.
$P$ is separable if and only if it has $n$ distinct roots in every field extension where $P$ splits.
A separable polynomial is also seen defined as a polynomial whose irreducible factors are separable in the sense of the definition above.