Definition:P-adic Norm
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Definition
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The $p$-adic norm is a norm on the set of rational numbers which yields a different topology from the regular Euclidean Metric.
Consider the $p$-adic valuation $\nu_p:\Q\to\Z\cup\{+\infty\}$
Define a map $|\cdot|_p:\Q \to \R_+$ as
- $|x|_p = \dfrac 1 {p^{\nu_p(x)}} $
Considering that $\dfrac 1 {p^{+\infty}}$ is defined $\dfrac 1 {p^{+\infty}}=0$ as it would be natural.
$|\cdot|_p$ forms a norm on the rational numbers, as is proved in P-adic Norm is a Norm which induces a metric by
- $d(x,y) = |x-y|_p$
