Definition:P-adic Metric

Definition

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

Rational Numbers

Let $\norm {\cdot}_p: \Q \to \R_{\ge 0}$ be the $p$-adic norm on $\Q$.

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

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

$p$-adic Numbers

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 $\Q$.