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'complete ultrametric field'
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| Title of object: |
complete ultrametric field |
| Canonical Name: |
CompleteUltrametricField |
| Type: |
Theorem |
| Created on: |
2005-01-03 14:29:00 |
| Modified on: |
2005-01-03 14:29:00 |
| Classification: |
msc:54E35, msc:12J10 |
| Keywords: |
convergence |
| Defines: |
ultrametric field, non-archimedean field |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
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 an ultrametric field. \,A necessary and sufficient condition for the convergence of the series \,$a_1+a_2+...$\, in $K$ is that
$$\lim_{n\to\infty}a_n = 0.$$ |
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