### 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 $\mathbb{Z}/(p)$ used in modular arithmetic, 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 $0 \leq a_n \lt p$, added and multiplied with the usual method of carrying familiar from adding and multiplying ordinary integers.

2. More abstractly, it is the limit $\underset{\leftarrow}{\lim} \mathbb{Z}/(p^n)$, in the category of (unital) rings, of the diagram

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

This is also a limit in the category of topological rings, taking the rings in the diagram to have discrete topologies.

3. Alternatively, it is the metric 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.

4. Also $\mathbb{Z}[ [ x ] ]/(x-q)\mathbb{Z}[ [ x ] ]$, see at analytic completion.

Hence one also speaks of the $p$-adic completion of the integers. See completion of a ring (which generalizes 2&3).

## Properties

### Topology

The ring of $p$-adic integers has the following properties:

### Relation to profinite completion of the integers

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

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.

### As the formal neighbourhood of a prime

The formal spectrum $Spf(\mathbb{Z}_p)$ of $\mathbb{Z}_p$ may be understood as the formal neighbourhood of the point corresponding to the prime $p$ in the prime spectrum $Spec(\mathbb{Z})$ of the integers. The inclusion

$\{p\} \hookrightarrow Spf(\mathbb{Z}_p) \hookrightarrow Spec(\mathbb{Z})$

is the formal dual of the canonical projection maps $\mathbb{Z}\to \mathbb{Z}_p\to \mathbb{Z}/(p)$.

## 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 April 11, 2014 11:30:43 by Urs Schreiber (89.204.135.175)