Semilocal units become principal after a fixed power: $u^{M} \equiv 1 \pmod{\mathfrak{p}}$ for all $\mathfrak{p} \mid p$

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Semilocal units become principal after a fixed power: $u^{M} \equiv 1 \pmod{\mathfrak{p}}$ for all $\mathfrak{p} \mid p$ · Prove2Me