# 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 $N(4)$.

$\blacksquare$