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


 * $$ |0|_p = 0, \,$$
 * $$ \left|p^n.\frac rs\right|_p = p^{-n} . \, $$

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.