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# $p$-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

$\xymatrix{&\mathbb{Z}\ar[dl]\ar[dr]\\ \mathbb{Z}/p^{{n+1}}\mathbb{Z}\ar[rr]&&\mathbb{Z}/p^{n}\mathbb{Z}}$ |

commutes. An algebraic and topological isomorphism between the two constructions is obtained by taking the coordinatewise projection map $\mathbb{Z}\to\,\underset{\longleftarrow}{\lim}\,\mathbb{Z}/p^{n}\mathbb{Z}$, extended to the completion of $\mathbb{Z}$ under the $p$–adic metric.

This alternate characterization shows that $\mathbb{Z}_{p}$ is compact, since it is a closed subspace of the space

$\prod_{{n\geq 0}}\mathbb{Z}/p^{n}\mathbb{Z}$ |

which is an infinite product of finite topological spaces and hence compact under the product topology.

# 3 Generalizations

If we interpret the prime $p$ as an equivalence class of valuations on $\mathbb{Q}$, then the field $\mathbb{Q}_{p}$ is simply the completion of the topological field $\mathbb{Q}$ with respect to the metric induced by any member valuation of $p$ (indeed, the valuation defined in Equation (1), extended to $\mathbb{Q}$, may serve as the representative). This notion easily generalizes to other fields and valuations; namely, if $K$ is any field, and $\mathfrak{p}$ is any prime of $K$, then the $\mathfrak{p}$–adic field $K_{\mathfrak{p}}$ is defined to be the completion of $K$ with respect to any valuation in $\mathfrak{p}$. The analogue of the $p$–adic integers in this case can be obtained by taking the subset (and subring) of $K_{\mathfrak{p}}$ consisting of all elements of absolute value less than or equal to $1$, which is well defined independent of the choice of valuation representing $\mathfrak{p}$.

In the special case where $K$ is a number field, the $\mathfrak{p}$–adic ring $K_{\mathfrak{p}}$ is always a finite extension of $\mathbb{Q}_{p}$ whenever $\mathfrak{p}$ is a finite prime, and is always equal to either $\mathbb{R}$ or $\mathbb{C}$ whenever $\mathfrak{p}$ is an infinite prime.

## Mathematics Subject Classification

11S99*no label found*12J12

*no label found*

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Compactness by AxelBoldt ✓

p-adic integers by pahio ✓

completion by pahio ✘