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
From Leigh.Samphier/Sandbox/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 Corollary to Valuation Ideal is Maximal Ideal of Induced Valuation Ring:
 * $\Z_p$ is a local ring