Ostrowski’s valuation theorem

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

• the trivial valuation,

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

• 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  $d(x,y)=|x-y|$  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 $ℝ$ and the fields ${\mathbb{Q}}_p$ of $p$-adic numbers (http://planetmath.org/PAdicIntegers); cf. also $p$-adic canonical form (http://planetmath.org/PAdicCanonicalForm).