\begin{definition}[Definition:P-adic Valuation/P-adic Numbers/Definition 2]
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 mapping $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ defined by:
:$(1): \quad \map {\nu_p} 0 = +\infty$
:$(2): \quad $for all $x \in \Q_p \setminus \set 0$:
:::$\map {\nu_p} x$ is the index of the first non-zero coefficient in the canonical $p$-adic expansion of $x$
\end{definition}