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

$\blacksquare$

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.8$ Algebraic properties of $p$-adic integers: Proposition $1.44$