 Completely split small-prime obstruction to monogenicity

Let $K$ have degree $n$, and suppose a rational prime $p$ factors into $n$ distinct prime ideals of norm $p$. If $n>p$, then $K$ is not monogenic. Otherwise the Dedekind factorization theorem would force the degree-$n$ minimal polynomial of a power-basis generator to have $n$ distinct linear factors over ${\mathbb{F}}_p$, although only the $p$ polynomials $X-a$ are available.