= Finiteness of S-unramified Kummer classes
{title2=$\#K(S,n)<\infty$}
The <S-unramified power class group> of a <number field> is finite. For $[a]\in K(S,n)$, the fractional ideal of $a$ away from $S$ is an $n$th power $\mathfrak a^n$. Sending $[a]$ to $[\mathfrak a]$ gives an exact sequence
$$
0\longrightarrow\mathcal O_{K,S}^{\times}/(\mathcal O_{K,S}^{\times})^n\longrightarrow K(S,n)\longrightarrow\operatorname{Cl}(\mathcal O_{K,S})[n]\longrightarrow0.
$$
The <S-unit group> is finitely generated, and the localized <ideal class group> is a quotient of the finite ordinary ideal class group. Both end groups are finite. This is the arithmetic finiteness needed in the <Kummer-theoretic proof of the weak Mordell-Weil theorem>.
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