P-adic Integers Form Integral Domain

Theorem
Let $\Q_p$ be the $p$-adic numbers for some prime $p$.

Let $\Z_p$ be the $p$-adic integers.

Then:
 * $\Z_p$ is an integral domain

Proof
From Field is Integral Domain:
 * $\Q_p$ is an integral domain

From P-adic Integers is Valuation Ring Induced by P-adic Norm/Corollary:
 * $\Z_p$ is a local ring

By the definition of local ring:
 * $\Z_p$ is a ring with unity

From Subdomain Test:
 * $\Z_p$ is an integral domain