P-adic Norm forms Non-Archimedean Valued Field

Theorem

Rational Numbers

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.

$p$-adic Numbers

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Then:

$\struct{\Q_p, \norm {\,\cdot\,}_p}$ is a valued field

$\norm {\,\cdot\,}_p$ is a non-Archimedean norm

That is, the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a valued field with a non-Archimedean norm.