P-adic Norm satisfies Non-Archimedean Norm Axioms

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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 $N(4)$.

$\blacksquare$