P-adic metric

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.

Definition

Every non-zero rational number may be written uniquely in the form $p^{n}.r/s$ where r and s are integers coprime to p and n is an integer. We define the p-adic valuation $|\cdot |_{p}$ on Q by

$|0|_{p}=0,\,$

$\left|p^{n}.{\frac {r}{s}}\right|_{p}=p^{-n}.\,$

The p-adic metric is then defined by

$d_{p}(x,y)=|x-y|_{p}.\,$

Properties

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.