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Revision difference : complete ultrametric field
Version 2 Version 1
A field $K$ together with its ultrametric $d$ is an {\em ultrametric field}; we can also speak of a {\em non-archimedean field} if the ultrametric is induced by a \PMlinkname{non-archimedean}{KrullValuation} valuation of $K$. A field $K$ together with its ultrametric $d$ is an {\em ultrametric field}; we can also speak of a {\em non-archimedean field} if the ultrametric is induced by a \PMlinkname{non-archimedean}{KrullValuation} valuation of $K$.
\textbf{Theorem.} \,Let $(K,\,d)$ be a complete ultrametric field. \,A necessary and sufficient condition for the convergence of the {\em series} \,$a_1+a_2+...$\, in $K$ is that \textbf{Theorem.} \,Let $(K,\,d)$ be an ultrametric field. \,A necessary and sufficient condition for the convergence of the series \,$a_1+a_2+...$\, in $K$ is that
$$\lim_{n\to\infty}a_n = 0.$$ $$\lim_{n\to\infty}a_n = 0.$$