 Solution

For any prime $p$, takeThe Eisenstein criterion, applied at the prime element $t$ of ${\mathbb{F}}_p[t]$, shows that $f$ is irreducible over $K$ and therefore has degree $p$. If $\alpha$ is a root in any extension field, the Frobenius endomorphism givesIt follows that this purely inseparable polynomial over a rational function field has at most one distinct root in every extension, and that root has multiplicity $p$, as required.