Definition:P-adic Valuation/P-adic Numbers/Definition 2

Definition
Let $p$ be a prime number.

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:
 * (1) $\map {\nu_p} 0 = +\infty$
 * (2) 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

 * Leigh.Samphier/Sandbox/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 $\Q_p$.