# P-adic Integers is Valuation Ring Induced by P-adic Norm/Corollary

## Theorem

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

Then:

the $p$-adic integers, $\Z_p$, is a local ring

## Proof

$\Z_p$ is the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$
$\Z_p$ is a local ring

$\blacksquare$