Definition:P-adic Integer/Definition 1

Definition
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

An element $x \in \Q_p$ is called a $p$-adic integer $\norm x_p \le 1$.

The set of all $p$-adic integers is usually denoted $\Z_p$.

Thus:
 * $\Z_p = \set {x \in \Q_p: \norm x_p \le 1}$

Also see

 * Leigh.Samphier/Sandbox/P-adic Integers is Valuation Ring Induced by P-adic Norm


 * Corollary to Valuation Ideal is Maximal Ideal of Induced Valuation Ring for a proof that the set of $p$-adic integers $\Z_p$ is a local ring.