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:
 * $(a): \quad$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
From P-adic Integers is Valuation Ring Induced by P-adic Norm:
 * $\Z_p$ is the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$

From Valuation Ideal of P-adic Numbers:
 * $p\Z_p$ is the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$

From Corollary to Valuation Ideal is Maximal Ideal of Induced Valuation Ring:
 * $(a):\quad \Z_p$ is a local ring
 * $(b):\quad p\Z_p$ is the unique maximal ideal of $\Z_p$