P-adic metric

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The p -adic metric, with respect to a given prime number p, on the field Q of rational numbers is a metric which is a valuation on the field.


Every non-zero rational number may be written uniquely in the form where r and s are integers coprime to p and n is an integer. We define the p-adic valuation on Q by

The p-adic metric is then defined by


The p-adic metric on Q is not complete: the p-adic numbers are the corresponding completion.

Ostrowksi's Theorem

The p-adic metrics and the usual absolute value on Q are mutually inequivalent. Ostrowkski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.