P-adic Norm forms Non-Archimedean Valued Field/Rational Numbers

Theorem
The $p$-adic norm $\norm{\,\cdot\,}_p$ forms a non-Archimedean norm on the rational numbers $\Q$.

The rational numbers $\struct{\Q, \norm{\,\cdot\,}_p}$ with the $p$-adic norm is a valued field with a non-Archimedean norm.

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

Let $\nu_p$ denote the $p$-adic valuation on the rational numbers.

Recall the definition of the $p$-adic norm:


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

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

We must show the following holds for all $x, y \in \Q$:
 * $\norm {x + y}_p \le \max \set {\norm x_p, \norm y_p}$

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

Let $x = 0$.

Then:

and so $\norm {x + y}_p \le \max \left( \norm x_p, \norm y_p \right)$

The same argument holds for $y = 0$.

Let $x + y = 0$.

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

From $p$-adic Valuation is Valuation:


 * $\map {\nu_p} {x + y} \ge \min \set {\map {\nu_p} x, \map {\nu_p} y}$

Then: