Definition:P-adic Norm

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

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

The $p$-adic norm on $\Q$ is the mapping $\left\Vert{\cdot}\right\Vert_p: \Q \to \R_{\ge 0}$ defined as:


 * $\forall q \in \Q: \left\Vert{q}\right\Vert_p := \begin{cases}

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

Also see

 * $p$-adic Norm is Norm

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