nLab
complete field

Context

Analysis

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Basic concepts

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Basic facts

Theorems

Algebra

Algebraic theories

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Algebras and modules

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Higher algebras

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

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Model category presentations

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Geometry on formal duals of algebras

Theorems

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 kk, there is a topology on kk 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 kk a field complete with respect to an absolute value ||{\vert - \vert} either the absolute value is archimedean, in which case kk 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.