Definition:P-adic Integer

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

The $p$-adic integers, denoted $\mathbf {Z}_p$, is the valuation ring induced by $\norm {\,\cdot\,}_p$, that is:
 * $\mathbf {Z}_p = \set{x \in \Q_p : \norm{x}_p \le 1}$

Note
In other sources the $p$-adic integers are often denoted $\Z_p$, but this notation is also used for the ring of integers module m where $m$ is a prime number. So has adopted the notation $\mathbf {Z}_p$.