Definition:P-adic Norm/Rational Numbers

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

Also see

 * Equivalence of Definitions of P-adic Norms


 * $p$-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 $\set {p^n : n \in \Z} \cup \set 0$