# $p$-adic integers

## 1 Basic construction

For any prime $p$, the $p$–adic integers is the ring obtained by taking the completion of the integers $\mathbb{Z}$ with respect to the metric induced by the norm

 $|x|:=\frac{1}{p^{\nu_{p}(x)}},\ \ x\in\mathbb{Z},$ (1)

where $\nu_{p}(x)$ denotes the largest integer $e$ such that $p^{e}$ divides $x$. The induced metric $d(x,y):=|x-y|$ is called the $p$–adic metric on $\mathbb{Z}$. The ring of $p$–adic integers is usually denoted by $\mathbb{Z}_{p}$, and its fraction field by $\mathbb{Q}_{p}$.

## 2 Profinite viewpoint

The ring $\mathbb{Z}_{p}$ of $p$–adic integers can also be constructed by taking the inverse limit

 $\mathbb{Z}_{p}:=\,\underset{\longleftarrow}{\lim}\,\mathbb{Z}/p^{n}\mathbb{Z}$

over the inverse system $\cdots\to\mathbb{Z}/p^{2}\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}\to 0$ consisting of the rings $\mathbb{Z}/p^{n}\mathbb{Z}$, for all $n\geq 0$, with the projection maps defined to be the unique maps such that the diagram