nLab
complete field
Context
Analysis
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Basic facts

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Theorems

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Algebra

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Contents
Definition
A field (as an algebraic structure ) equipped with a metric is complete if the operations are continuous with respect to the induced topology and it is a complete metric space .

The most important case is the case of valued fields (i.e. multiplicatively normed fields ).

Given a multiplicative discrete valuation on a field $k$ , there is a topology on $k$ induced by the metrics induced by the valuation . A complete valued field is a field complete with respect to the valuation metric.

Properties
One of Ostrowski's theorems says that for $k$ a field complete with respect to an absolute value ${\vert - \vert}$ either the absolute value is archimedean, in which case $k$ is either the field of real numbers or of complex numbers , or the absolute value is non-archimedean.

(submultiplicative)
multiplicative (/)

Last revised on July 13, 2014 at 07:55:12.
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