### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

For any prime number $p$, the ring of $p$-adic integers ${ℤ}_{p}$ (which, to avoid possible confusion with the ring $ℤ/\left(p\right)$, is also written as ${\stackrel{^}{ℤ}}_{p}$) 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

$\sum _{n\ge 0}{a}_{n}{p}^{n}$\sum_{n \geq 0} a_n p^n

where $0\le {a}_{n}\le p-1$, 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 $ℤ$ 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 ${ℤ}_{p}$. Thus ${ℤ}_{p}$ is a topological ring.

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

$\dots \to ℤ/\left({p}^{n+1}\right)\to ℤ/\left({p}^{n}\right)\to \dots \to ℤ/\left(p\right)$\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

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

Revised on August 14, 2012 01:24:17 by Toby Bartels (98.19.44.121)