# Confused about separable elements with respect to a field

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 $p$.
• The minimal polynomial of $t$ is $X^p-t^p$, which is irreducible over $K(t^p)$ due to Eisenstein. Now you should check that this polynomial is obviously not separable. – MooS Feb 5 '15 at 16:51
• I see that $X^p-t^p=(X-t)^p$, so it's not separable. But can you explain how you obtained the minimal polynomial? – odnerpmocon Feb 5 '15 at 16:56