Definition:P-adic Norm

Definition
The $p$-adic norm is a norm on the set of rational numbers which yields a different topology from the regular Euclidean Metric.

Consider the $p$-adic valuation $\nu_p:\Q\to\Z\cup\{+\infty\}$

Define a map $|\cdot|_p:\Q \to \R_+$ as


 * $|x|_p = \dfrac 1 {p^{\nu_p(x)}} $

Considering that $\dfrac 1 {p^{+\infty}}$ is defined $\dfrac 1 {p^{+\infty}}=0$ as it would be natural.

$|\cdot|_p$ forms a norm on the rational numbers, as is proved in P-adic Norm is a Norm which induces a metric by


 * $d(x,y) = |x-y|_p$