= Unramified Kummer classes with bounded prime support
{title2=$K(S,m)=\{[a]:v_{\mathfrak p}(a)\equiv0\pmod m\text{ outside }S\}$}
For a <number field> $K$, a finite set $S$ and $m\geq2$, let $K(S,m)$ consist of classes $[a]\in K^*/K^{*m}$ whose valuations outside $S$ are divisible by $m$. It fits into $0\to\mathcal O_{K,S}^{\times}/(\mathcal O_{K,S}^{\times})^m\to K(S,m)\to\operatorname{Cl}(\mathcal O_{K,S})[m]\to0$. The <S-unit group> is finitely generated and the <ideal class group> is finite, so $K(S,m)$ is finite. When $S$ contains primes dividing $m$, this controls unramified multiplicative Kummer classes outside $S$.
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