In van der Waerden's Algebra, a separable element with respect to a field $F$ is defined as the following: If $\alpha$ is a root of a polynomial irreducible in $F[X]$ which has only separated(simple) roots, then $\alpha$ is called separable.
Looking at the definition, I don't find it clear how and why some elements are not separable. Could someone give an example of a field and an inseparable element over it?
The element $t$ (transcendental over $K$) in the field extension $K(t)/K(t^p)$ is not separable, when $K$ has characteristic $p$.