nLab complete field

Definition

A field (as an algebraic structure) equipped with a metric is complete if its 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.

Related concepts

• archimedean field, archimedean valued field

algebraic structure	group	ring	field	vector space	algebra
(submultiplicative) norm	normed group	normed ring	normed field	normed vector space	normed algebra
multiplicative norm (absolute value/valuation)			valued field
completeness	complete normed group	Banach ring	complete field	Banach vector space	Banach algebra

References

See also:

• Wikipedia, Complete field