Definition:P-adic Norm/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 on Rational Numbers 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 $\set {p^n : n \in \Z} \cup \set 0$