Definition

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

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

$\map {\nu_p^\Z} n := \begin {cases} +\infty & : n = 0 \\ \sup \set {v \in \N: p^v \divides n} & : n \ne 0 \end{cases}$

where:

$\sup$ denotes supremum
$p^v \divides n$ 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.