# Definition:P-adic Norm/Rational Numbers

< Definition:P-adic Norm(Redirected from Definition:P-adic Norm on Rational Numbers)

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## Definition

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

### Definition 1

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:

- $\forall q \in \Q: \norm q_p := \begin{cases} 0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end{cases}$

### Definition 2

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

- $\forall r \in \Q: \norm r_p = \begin {cases} 0 & : r = 0 \\ \dfrac 1 {p^k} & : r = p^k \dfrac m n: k, m, n \in \Z, p \nmid m, n \end {cases}$

## Also see

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

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

- $p$-adic Norm and Absolute Value are Not Equivalent where it is shown that the
**$p$-adic norm**yields a different topology on the rational numbers 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.

- P-adic Norm Characterisation of Divisibility by Power of p where divisibilty by a power of $p$ is characterised by the $p$-adic norm.

- Image of P-adic Norm where it is shown that the image of $\norm {\,\cdot\,}_p$ is $\set {p^n : n \in \Z} \cup \set 0$