Definition:P-adic Norm

Definition
Let $p \in \N$ be a prime.

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

The $p$-adic norm on $\Q$ is the mapping $\left\vert{\cdot}\right\vert_p : \Q \to \R_{\ge 0}$ defined by:


 * $\left\vert{q}\right\vert_p := \begin{cases}

0 & : q = 0 \\ p^{- \nu_p (q)} & : q \ne 0 \end{cases}$

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

$p$-adic Metric
The $p$-adic metric on $\Q$ is the metric induced by $\left\vert{\cdot}\right\vert_p$.

Thus, for all $x, y \in \Q$, it is defined by:


 * $d \left({x,y}\right) = \left\vert{x - y}\right\vert_p$

Also see

 * P-adic Norm is a Norm
 * Ostrowski's Theorem