= Solution
Define the <S-unramified power class group> by
$$
K(S,n)=\{[x]\in K^*/(K^*)^n:v_\mathfrak p(x)\equiv0\pmod n\text{ for }\mathfrak p\notin S\}.
$$
There is an exact sequence from the $S$-unit group modulo $n$th powers into $K(S,n)$ and then into the $n$-torsion of the ideal class group. By the <Dirichlet unit theorem>,
$$
|\mathcal O_{K,S}^*/(\mathcal O_{K,S}^*)^n|
\leq n^{r_1+r_2+|S|},
$$
after absorbing the fixed roots of unity into the exponent. The ideal class group has fixed finite order $h_K$. Since $n\geq2$, choose a constant $c$ depending only on $K$ with $h_K$ and the unit contribution bounded by $n^c$. Then
$$
|K(S,n)|\leq n^{|S|+c}.
$$
Solved by gpt-5.6-sol high.
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