P-adic Norm satisfies Non-Archimedean Norm Axioms

Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers with $p$-adic norm $\norm {\,\cdot\,}_p : \Q_p \times \Q_p \to \R_{\ge 0}$.

Then $\norm {\,\cdot\,}_p$ satisfies the non-Archimedean norm axioms:

\((N1)\)	$:$	Positive Definiteness:	\(\displaystyle \forall x \in \Q_p:\)		\(\displaystyle \norm x_p = 0 \)	\(\displaystyle \iff \)	\(\displaystyle x = 0 \)
\((N2)\)	$:$	Multiplicativity:	\(\displaystyle \forall x, y \in \Q_p:\)		\(\displaystyle \norm {x \cdot y}_p \)	\(\displaystyle = \)	\(\displaystyle \norm x_p \times \norm y_p \)
\((N4)\)	$:$	Ultrametric Inequality:	\(\displaystyle \forall x, y \in Q_p:\)		\(\displaystyle \norm {x + y}_p \)	\(\displaystyle \le \)	\(\displaystyle \max \set {\norm x_p, \norm y_p} \)

Proof

By definition, the $p$-adic numbers are the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$.

By definition of a non-Archimedean valued field, $\norm {\,\cdot\,}_p$ is a non-Archimedean norm.

By definition of a non-Archimedean norm, $\norm {\,\cdot\,}_p$ satisfies the axioms: $N(1)$, $N(2)$ and $N(4)$.

$\blacksquare$