Definition:P-adic Norm/Rational Numbers/Definition 1

Definition

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

Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$.

The $p$-adic norm on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as:

$\forall q \in \Q: \norm q_p := \begin{cases}

0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end{cases}$

Also see

• Equivalence of Definitions of P-adic Norms

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.1$: Absolute Values on a Field: Definition $2.1.4$
• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$