Ostrowski’s valuation theorem

The field of rational numbers has no other non-equivalent (http://planetmath.org/EquivalentValuations) valuationsMathworldPlanetmath than

Note.  Any valuation || of the field defines a metric  d(x,y)=|x-y|  in the field, but is completePlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Complete) only with respect to (the “trivial metric” defined by) the trivial valuation.  The field has the proper completions with respect to its other valuations:  the field of reals and the fields p of p-adic numbers (http://planetmath.org/PAdicIntegers); cf. also p-adic canonical formMathworldPlanetmath (http://planetmath.org/PAdicCanonicalForm).

Title Ostrowski’s valuation theorem
Canonical name OstrowskisValuationTheorem
Date of creation 2013-03-22 14:55:30
Last modified on 2013-03-22 14:55:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 13A18