Definition:P-adic Valuation/P-adic Numbers

Definition

Let $p$ be a prime number.

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

Definition 1

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}$

Definition 2

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

$(1): \quad \map {\nu_p} 0 = +\infty$

$(2): \quad $for all $x \in \Q_p \setminus \set 0$:

$\map {\nu_p} x$ is the index of the first non-zero coefficient in the canonical $p$-adic expansion of $x$

Also see

• Equivalence of Definitions of P-adic Valuation on P-adic Numbers

• P-adic Valuation Extends to P-adic Numbers where it is shown that $\nu_p$ is a valuation that extends the $p$-adic valuation on the rational numbers $\Q$ to $\Q_p$.