Ideals of P-adic Integers/Corollary

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Then:
 * $\Z_p$ is a principal ideal domain

Proof
From Ideals of P-adic Integers, all ideals of $\Z_p$ are the principal ideals:
 * $\quad\set 0$
 * $\quad\forall n \in \N : p^n \Z_p$

Hence $\Z_p$ is a principal ideal domain by definition.