Definition:P-adic Valuation

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

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


 * $\nu_p^\Z \left({n}\right) := \begin{cases}

+\infty & : n = 0 \\ \sup \left\{{v \in \N: p^v \mathbin \backslash n}\right\} & : n \ne 0 \end{cases}$

where $\sup$ denotes supremum, and $p^v \mathbin \backslash n$ expresses that $p^v$ divides $n$.

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


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

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

Also see

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