example of nonperfect field

In this entry, we exhibit an example of a field that is not a perfect field.

Let $F={𝔽}_p(t)$, where ${𝔽}_p$ is the field with $p$ elements and $t$ transcendental over ${𝔽}_p$. The splitting field $E$ of the irreducible polynomial $f=x^p-t$ is not separable over $F$. Indeed, if $\alpha$ is an element of $E$ such that ${\alpha }^p=t$, we have

$$x^p-t=x^p-{\alpha }^p={(x-\alpha )}^p,$$

which shows that $f$ has one root of multiplicity $p$.