# nLab p-adic integer

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

For any prime number $p$, the ring of $p$-adic integers $\mathbb{Z}_p$ (which, to avoid possible confusion with the ring $\mathbb{Z}/(p)$, is also written as $\widehat{\mathbb{Z}}_p$) may be described in one of several ways:

1. To the person on the street, it may be described as (the ring of) numbers written in base $p$, but allowing infinite expansions to the left. Thus, numbers of the form

$\sum_{n \geq 0} a_n p^n$

where $0 \leq a_n \leq p-1$, added and multiplied with the usual method of carrying familiar from adding and multiplying ordinary integers.

2. More precisely, it is the metric space completion of the ring of integers $\mathbb{Z}$ with respect to the $p$-adic absolute value. Since addition and multiplication of integers are uniformly continuous with respect to the $p$-adic absolute value, they extend uniquely to a uniformly continuous addition and multiplication on $\mathbb{Z}_p$. Thus $\mathbb{Z}_p$ is a topological ring.

3. Alternatively, it is the limit, in the category of (unital) rings, of the diagram

$\ldots \to \mathbb{Z}/(p^{n+1}) \to \mathbb{Z}/(p^n) \to \ldots \to \mathbb{Z}/(p)$

also considered as a topological ring if the limit is taken in the category of topological rings, and taking the rings in the diagram to have discrete topologies.

## Properties

### Topology

The $p$-adic integers have the following properties:

### Relation to profinite completion of the integers

###### Example

The profinite completion of the integers is

$\widehat {\mathbb{Z}} \coloneqq \underset{\leftarrow}{\lim}_{n \in \mathbb{N}} (\mathbb{Z}/n\mathbb{Z}) \,.$

This is isomorphic to the product of the $p$-adic integers for all $p$

$\widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p \,.$
###### Definition

The ring of integral adeles $\mathbb{A}_{\mathbb{Z}}$ is the product of the profinite completion $\widehat{\mathbb{Z}}$ of the integers, example 1, with the real numbers

$\mathbb{A}_{\mathbb{Z}} \coloneqq \mathbb{R} \times \widehat{\mathbb{Z}} \,.$

The group of units of the ring of adeles is called the group of ideles.

## References

Introductions and surveys include

• Bernard Le Stum, One century of $p$-adic geometry – From Hensel to Berkovich and beyond talk notes, June 2012 (pdf)

• Hendrik Lenstra, Profinite groups (pdf)

Revised on November 21, 2013 12:04:38 by Urs Schreiber (188.200.54.65)