\begin{definition}[Definition:P-adic Valuation/Integers]
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 $n$.
\end{definition}