## Theorem

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

Then:

$(a): \quad$the $p$-adic integers, $\Z_p$, is a local ring
$(b): \quad$the principal ideal $p\Z_p$ is the unique maximal ideal of $\Z_p$

## Proof

$\Z_p$ is the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$
$p\Z_p$ is the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$
$(a):\quad \Z_p$ is a local ring
$(b):\quad p\Z_p$ is the unique maximal ideal of $\Z_p$

$\blacksquare$