p-adic integer

Remarks

By the strong triangle inequality, $\Z_p$ is a subring of $\Q_p$. Its units are the $x$ with $\abs{x}_p = 1$, and its only nonzero prime ideal is $p \Z_p$. Equivalently, $\Z_p$ is the set of sums $\sum_{n \ge 0} a_n p^n$ with digits $a_n \in \set{0, \ldots, p - 1}$. Equivalently, $\Z_p$ is the inverse limit of the rings $\Z / p^n \Z$ under reduction maps. Example: $1/2 \in \Z_3$, because $3$ does not divide $2$, so $\abs{1/2}_3 = 1$. Non-example: $1/3 \notin \Z_3$, because $\abs{1/3}_3 = 3 > 1$. A rational number lies in $\Z_p$ exactly when its reduced denominator is not divisible by $p$.