= Solution
For any prime $p$, take
$$
K=\mathbb F_p(t),
\qquad
f(X)=X^p-t.
$$
The <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> gives
$$
f(X)=X^p-\alpha^p=(X-\alpha)^p.
$$
It 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.
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