Ostrowski’s valuation theorem
The field of rational numbers has no other nonequivalent (http://planetmath.org/EquivalentValuations) valuations^{} than

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the trivial valuation,

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the absolute value^{}, i.e. the complex modulus $\cdot {}_{\mathrm{\infty}}$ and

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the $p$adic valuations $\cdot {}_{p}$ when $p$ goes through all positive primes.
Note. Any valuation $\cdot $ of the field $\mathbb{Q}$ defines a metric $d(x,y)=xy$ in the field, but $\mathbb{Q}$ is complete^{} (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 $\mathbb{R}$ and the fields ${\mathbb{Q}}_{p}$ of $p$adic numbers (http://planetmath.org/PAdicIntegers); cf. also $p$adic canonical form^{} (http://planetmath.org/PAdicCanonicalForm).
Title  Ostrowski’s valuation theorem 

Canonical name  OstrowskisValuationTheorem 
Date of creation  20130322 14:55:30 
Last modified on  20130322 14:55:30 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  9 
Author  pahio (2872) 
Entry type  Theorem 
Classification  msc 13A18 