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'independence of valuations'
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| Title of object: |
independence of valuations |
| Canonical Name: |
IndependenceOfTheValuations |
| Type: |
Theorem |
| Created on: |
2004-02-25 17:59:09 |
| Modified on: |
2004-10-31 09:09:13 |
| Classification: |
msc:11R99 |
| Synonyms: |
independence of valuations=approximation theorem |
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}
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%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
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%\usepackage{xypic}
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% define commands here |
Content:
Let $|\cdot|_1$, \ldots, $|\cdot|_n$ be {\em non-trivial} (i.e., they all have also other values than 0 and 1) and pairwise non-equivalent valuations of a field $K$. If $a_1$, ..., $a_n$ are some elements of this field and $\varepsilon$ is an arbitrary positive number, then there exists in $K$ an element $y$ which satisfies the conditions
\begin{align*}
\begin{cases}
|y-a_1|_1 < \varepsilon,\\
\qquad \vdots \qquad \\
|y-a_n|_n < \varepsilon.\\
\end{cases}
\end{align*} |
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