Definition:P-adic Integer

Definition
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the $p$-adic numbers $\Q_p$ 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:
 * $\Z_p = \set {x \in \Q_p: \norm x_p \le 1}$

Notation
The notation $\Z_p$ is also used for the ring of integers module $p$ where $p$ is a prime number.

On the context of any page where $\Z_p$ appears will define what is referred to by $\Z_p$.

Also see

 * 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.