Definition:P-adic Norm

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

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

The $p$-adic norm on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as:


 * $\forall q \in \Q: \norm q_p := \begin{cases}

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

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

Also see

 * $p$-adic Norm is Norm where it is shown that the $p$-adic norm is a norm on a division ring.


 * $p$-adic Norm is non-Archimedean Norm where it is shown that the $p$-adic norm is a non-Archimedean norm on a division ring.


 * $p$-adic Norm and Absolute Value are Not Equivalent where it is shown that the $p$-adic norm yields a different topology on the rationals from the usual Euclidean Metric.


 * $p$-adic Norms are Not Equivalent where it is shown that the $p$-adic norms for two distinct prime numbers are not equivalent norms.