Definition:P-adic Metric/P-adic Numbers

Definition

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

Let $\struct{\Q_p,\norm{\,\cdot\,}_p}$ be the $p$-adic numbers.

The $p$-adic metric on $\Q_p$ is the metric induced by $\norm{\,\cdot\,}_p$:

$\forall x, y \in \Q_p: \map d {x, y} = \norm{x - y}_p$

Also see

• Metric on P-adic Numbers Extends Metric on Rationals where it is shown that the $p$-adic metric on $\Q_p$ extends the $p$-adic metric on the rational numbers $\Q$.

• P-adic Metric on P-adic Numbers is Non-Archimedean Metric where is it is shown that the $p$-adic metric is a non-Archimedean metric.