# $p$-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^{}

$${\mathbb{Z}}_{p}:=\underset{\u27f5}{lim}\mathbb{Z}/{p}^{n}\mathbb{Z}$$ |

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