# example of nonperfect field

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

Let $F=\mathbb{F}_{p}(t)$, where $\mathbb{F}_{p}$ is the field with $p$ elements and $t$ transcendental over $\mathbb{F}_{p}$. 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 $p$.

Title example of nonperfect field ExampleOfNonperfectField 2013-03-22 13:08:31 2013-03-22 13:08:31 CWoo (3771) CWoo (3771) 7 CWoo (3771) Example msc 12F10