Definition:P-adic Valuation

Definition

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

The $p$-adic valuation (on $\Z$) is the mapping $\nu_p^\Z: \Z \to \N \cup \set {+\infty}$ defined as:

$\map {\nu_p^\Z} n := \begin {cases} +\infty & : n = 0 \\ \sup \set {v \in \N: p^v \divides n} & : n \ne 0 \end{cases}$

where:

$\sup$ denotes supremum

$p^v \divides n$ expresses that $p^v$ divides $n$.

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 $\nu_p: \Q \to \Z \cup \set {+\infty}$.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

The $p$-adic valuation on $p$-adic numbers is the function $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ defined by:

$\forall x \in \Q_p : \map {\nu_p} x = \begin {cases} -\log_p \norm x_p : x \ne 0 \\ +\infty : x = 0 \end {cases}$

The $p$-adic valuation is also known as the $p$-adic order.