\begin{definition}[Definition:P-adic Valuation/Rational Numbers]
Let $p \in \N$ be a prime number.
Let the $p$-adic valuation on the integers $\nu_p^\Z$ be extended to $\nu_p^\Q: \Q \to \Z \cup \set {+\infty}$ by:
:$\map {\nu_p^\Q} {\dfrac a b} := \map {\nu_p^\Z} a - \map {\nu_p^\Z} b$
This mapping $\nu_p^\Q$ is called the '''$p$-adic valuation (on $\Q$)''' and is usually denoted $\nu_p: \Q \to \Z \cup \set {+\infty}$.
\end{definition}