Definition:P-adic Valuation/P-adic Numbers/Definition 1
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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:
- $\forall x \in \Q_p : \map {\nu_p} x = \begin {cases}
-\log_p \norm x_p : x \ne 0 \\ +\infty : x = 0 \end {cases}$
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 $\Q_p$.
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$