 Local compactness criterion for a complete non-Archimedean field

A complete field with a nontrivial Non-Archimedean absolute value is locally compact exactly when its valuation is discrete and its residue field is finite. In that case its valuation ring is the inverse limit of the finite rings $\mathcal{O}/{\mathfrak{m}}^n$ and is compact. The nontrivial valuation in the local compactness criterion is essential.