Definition:P-adic Norm

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Definition

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

Rational Numbers

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:

$\forall q \in \Q: \norm q_p := \begin {cases} 0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end {cases}$

$p$-adic Numbers

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

Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by:

$\ds \forall \eqclass{x_n}{} \in \Q_p: \norm {\eqclass{x_n}{} }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

The norm $\norm {\,\cdot\,}_p$ on $\Q_p$ is called the $p$-adic norm on $\Q_p$.

Also see

• Rational Numbers are Dense Subfield of $p$-adic Numbers for a proof that the $p$-adic norm on $p$-adic numbers may be considered an extension of the $p$-adic norm on the rational numbers

• Results about $p$-adic norms can be found here.