$p$-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

$${\mathbb{Z}}_p:=\underset{⟵}{lim}\mathbb{Z}/p^n\mathbb{Z}$$

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