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Ostrowski’s valuation theorem
The field of rational numbers has no other nonequivalent valuations than

the trivial valuation,

the absolute value, i.e. the complex modulus $\cdot_{\infty}$ and
Note. Any valuation $\cdot$ of the field $\mathbb{Q}$ defines a metric $d(x,\,y)=xy$ in the field, but $\mathbb{Q}$ is 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; cf. also $p$adic canonical form.
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