complete field

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.

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.

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.

Revised on January 9, 2012 01:51:25
by Urs Schreiber
(82.113.119.184)