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

## Theorem

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.

## Proof

Let $\norm {\,\cdot\,}^\Q_p$ be the p-adic norm on the rationals $\Q$.

$\struct{Q, \norm {\,\cdot\,}^\Q_p}$ is a valued field with non-Archimedean norm $\norm {\,\cdot\,}_p$

By definition of the $p$-adic numbers:

$\Q_p$ is the quotient ring $\CC \, \big / \NN$

where:

$\CC$ is the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.

and

$\NN$ is the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.
$\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a valued field.
$\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a normed division ring completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$
$\norm {\, \cdot \,}_p$ on $\Q_p$ is a non-Archimedean norm.

$\blacksquare$