P-adic Integers Form Integral Domain

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

Then:
 * $\Z_p$ is an integral domain

Proof
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.

It follows that $\Z_p$ is a non-null ring.

It remains to show that $\Z_p$ has no non-zero divisors.

Let $a, b \in \Z_p \setminus \set 0$.

From P-adic Integer is Limit of Unique P-adic Expansion, let the $p$-adic expansions of $a, b$ and $ab$ be:
 * $a = \ds \sum_{n = 0}^\infty a_n p^n$
 * $b = \ds \sum_{n = 0}^\infty b_n p^n$
 * $ab = \ds \sum_{n = 0}^\infty c_n p^n$

Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ denote the $p$-adic valuation on the $p$-adic numbers.

Let:
 * $v = \map \nu a \ne 0$

and
 * $w = \map \nu b \ne 0$

By definition of valuation:
 * $\map \nu {ab} = v + w$

From :
 * $c_{v+w}$ is the first non-zero coefficient of $ab$

It follows that $ab \ne 0$.

The result follows.