Definition:P-adic Valuation/P-adic Numbers
< Definition:P-adic Valuation(Redirected from Definition:P-adic Valuation on P-adic Numbers)
Jump to navigation
Jump to search
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
- 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 $\Q_p$.