Definition:P-adic Valuation

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

Consider the mapping $\nu_p^\Z: \Z \to \N \cup \left\{{+\infty}\right\}$ such that:


 * $\nu_p^\Z \left({0}\right) = +\infty$
 * $\nu_p^\Z \left({n}\right)$ is the highest exponent $v \in \N$ such that $p^v$ divides $n$

Next, extend it to $\nu_p^\Q: \Q \to \Z \cup \left\{{+\infty}\right\}$ by:


 * $\nu_p^\Q \left(\dfrac{a}{b}\right) = \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right)$

$\nu_p^\Q$ is called the $p$-adic valuation (on $\Q$) and is usually denoted $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$.