 Solution

If $I=0$, take $n=1$. Otherwise, by part (b), the element $[I]$ of the finite group $\mathrm{Cl}(K)$ has finite order, say $n\ge 1$. Thereforewhich says precisely that $I^n$ is principal. Since $I^n$ is integral, if $I^n=(\alpha )$ then necessarily $\alpha \in {\mathcal{O}}_K$.