Definition:P-adic Valuation/Integers

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Definition

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


The $p$-adic valuation (on $\Z$) is the mapping $\nu_p^\Z: \Z \to \N \cup \left\{{+\infty}\right\}$ defined by:

$\nu_p^\Z \left({n}\right) := \begin{cases} +\infty & : n = 0 \\ \sup \left\{{v \in \N: p^v \mathbin \backslash n}\right\} & : n \ne 0 \end{cases}$

where:

$\sup$ denotes supremum
$p^v \mathbin \backslash n$ expresses that $p^v$ divides $n$.


Also known as

As the $p$-adic valuation is usually defined as its extension $\nu_p^\Q$ to the rationals, the $p$-adic valuation on $\Z$ is often seen referred to as the restricted $p$-adic valuation.


Also see