\begin{definition}[Definition:P-adic Valuation/P-adic Numbers/Definition 1]
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
The '''$p$-adic valuation on $p$-adic numbers''' is the function $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ defined by:
:$\forall x \in \Q_p : \map {\nu_p} x = \begin {cases}
-\log_p \norm x_p : x \ne 0 \\
+\infty : x = 0
\end {cases}$
\end{definition}