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 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.


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.

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