nLab
p-adic integer

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 ${\hat{\mathbb{Z}}}_p$) 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

   $$\sum _{n\ge 0}{a}_n{p}^n$$\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 $\mathbb{Z}$ 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

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

• As a topological space, it is compact, Hausdorff, and totally disconnected (i.e., is a Stone space). Moreover, every point is an accumulation point?, and there is a countable basis of clopen sets – a Stone space with these properties must be homeomorphic to Cantor space.

• As a topological group under addition, it is therefore an almost connected group. As an abelian compact group, it is Pontryagin dual to the Prüfer $p$-group as discrete group.

Related notions

• $p$-adic number, adele?.

• Z-infinity-module