Question : Let $\nu$ be a non-archimedean valuation on a number field $k$. Is $\mathcal{O}_{\nu}^{*}$ ( the set of units in ring of integers) compact under subspace topology of $k_{\nu}$ (completion of $k$ under $\nu$)?
I have proven that it is locally compact but it is not apparently clear to me whether it is compact or not.