P-adic Norm is Non-Archimedean Norm

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Theorem

The $p$-adic norm forms a non-Archimedean norm on the rational numbers $\Q$.


Proof

First we note that the $p$-adic norm is a norm.

Let $v_p$ be the $p$-adic valuation on the rational numbers.

Recall that the $p$-adic norm is defined as:

$\forall q \in \Q: \left\Vert{q}\right\Vert_p := \begin{cases} 0 & : q = 0 \\ p^{- \nu_p \left({q}\right)} & : q \ne 0 \end{cases}$

We must show the following holds for all $x, y \in \Q$:

$\left\Vert {x + y} \right\Vert_p \le \max \left\{ {\left\Vert {x} \right\Vert_p, \left\Vert {y} \right\Vert_p} \right\}$


If $x = 0$ or $y = 0$, or $x + y = 0$, the result is trivial, as follows:

Let $x = 0$.

Then:

\(\displaystyle x\) \(=\) \(\displaystyle 0\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left\Vert{x}\right\Vert_p\) \(=\) \(\displaystyle 0\) Definition of $p$-adic Norm
\(\displaystyle \implies \ \ \) \(\displaystyle \max \left( \left\Vert{x}\right\Vert_p, \left\Vert{y}\right\Vert_p \right)\) \(=\) \(\displaystyle \left\Vert{y}\right\Vert_p\) as $\left\Vert{y}\right\Vert_p \ge 0 = \left\Vert{x}\right\Vert_p$ from Norm Axioms: Axiom $(N1)$
\(\displaystyle \) \(=\) \(\displaystyle \left\Vert{x + y}\right\Vert_p\)

and so $\left\Vert{x + y}\right\Vert_p \le \max \left( \left\Vert{x}\right\Vert_p, \left\Vert{y}\right\Vert_p \right)$

The same argument holds for $y = 0$.


Let $x + y = 0$.

\(\displaystyle x + y\) \(=\) \(\displaystyle 0\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left\Vert{x + y}\right\Vert_p\) \(=\) \(\displaystyle 0\) Definition of $p$-adic Norm
\(\displaystyle \) \(\le\) \(\displaystyle \max \left( \left\Vert{x}\right\Vert_p, \left\Vert{y}\right\Vert_p \right)\) as $\left\Vert{x}\right\Vert_p \ge 0$ and $\left\Vert{y}\right\Vert_p \ge 0$ from Norm Axioms: Axiom $(N1)$


Let $x, y, x + y \in \Q_{\ne 0}$.

From $p$-adic Valuation is Valuation:

$\nu_p \left({x + y}\right) \ge \min \left\{ {\nu_p \left({x}\right), \nu_p \left({y}\right)}\right\}$

Then:

\(\displaystyle \left\Vert{x + y}\right\Vert_p\) \(=\) \(\displaystyle p^{- \nu_p \left({x + y}\right)}\) Definition of $p$-adic Norm
\(\displaystyle \) \(\le\) \(\displaystyle \max \left\{ {p^{- \nu_p \left({x}\right)}, p^{- \nu_p \left({y}\right)} }\right\}\)
\(\displaystyle \) \(=\) \(\displaystyle \max \left\{ {\left\Vert{x}\right\Vert_p, \left\Vert{y}\right\Vert_p}\right\}\) Definition of $p$-adic Norm

$\blacksquare$


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