# Definition:P-adic Norm

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## Definition

### Definition 1

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:

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

### Definition 2

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

Let $k, m, n \in \Z : p \nmid m, n$.

Let $\displaystyle r := p^k \frac m n$.

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

$\forall r \in \Q: \norm r_p := \begin{cases} 0 & : r = 0 \\ p^{-k} & : r \ne 0 \end{cases}$

### $p$-adic Metric

The $p$-adic metric on $\Q$ is the metric induced by $\norm{\cdot}_p$:

$\forall x, y \in \Q: \map d {x, y} = \norm{x - y}_p$

### $p$-adic Numbers

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

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

## Notes

The $p$-adic norm can only take values from the set $\set {p^n : n \in \Z} \cup \set{0}$

Also, if $a, b \in \Z$, then $a \equiv b \pmod {p^n}$ if and only if $\norm{a - b}_p \le p^{-n}$