Definition:P-adic Metric
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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 $\Q$.