nLab
complete field

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.

algebraic structuregroupringfieldvector spacealgebra
(submultiplicative) normnormed groupnormed ringnormed fieldnormed vector spacenormed algebra
multiplicative norm (absolute value/valuation)valued field
completenesscomplete normed groupBanach ringcomplete fieldBanach vector spaceBanach algebra

Revised on July 13, 2014 07:55:12 by Urs Schreiber (89.204.130.46)