 Solution

Taking norms ingivesso $s=n$.

Suppose for contradiction that ${\mathcal{O}}_K=\mathbb{Z}[\alpha ]$, and let $f$ be the degree-$n$ minimal polynomial of $\alpha$. Part (a) identifies the prime ideals above $p$ with the distinct irreducible factors of $\overline{f}$. Since there are $n$ distinct prime ideals, all with norm $p$, the norm formula forces $\overline{f}$ to have $n$ distinct irreducible factors of degree one. But ${\mathbb{F}}_p[X]$ has only the $p$ distinct monic linear polynomialsThis is impossible when $n>p$. The completely split small-prime obstruction to monogenicity therefore gives