Definition:P-adic Valuation/Rational Numbers

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

Let the $p$-adic valuation on the integers $\nu_p^\Z$ be extended to $\nu_p^\Q: \Q \to \Z \cup \left\{{+\infty}\right\}$ by:


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

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

Also see

 * P-adic Valuation of Rational Number is Well Defined
 * P-adic Valuation is Valuation, showing that indeed $\nu_p^\Q$ is a valuation