symmetric monoidal (∞,1)-category of spectra
Given a multiplicative discrete valuation on a field , there is a topology on 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 a field complete with respect to an absolute value either the absolute value is archimedean, in which case is either the field of real numbers or of complex numbers, or the absolute value is non-archimedean.
|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|