# nLab complete field

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## Higher algebras

• symmetric monoidal (∞,1)-category of spectra

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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. See the history of this page for a list of all contributions to it.