Local compactness criterion for a complete non-Archimedean field
= 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.