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p-adic integer

Context

Algebra

Formal geometry

Arithmetic geometry

Contents

Definition

For any prime number pp, the ring of pp-adic integers p\mathbb{Z}_p (which, to avoid possible confusion with the ring /(p)\mathbb{Z}/(p) used in modular arithmetic, is also written as ^ p\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 pp, but allowing infinite expansions to the left. Thus, numbers of the form

    n0a np n\sum_{n \geq 0} a_n p^n

    where 0a n<p0 \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 lim/(p n)\underset{\leftarrow}{\lim} \mathbb{Z}/(p^n), in the category of (unital) rings, of the diagram

    /(p n+1)/(p n)/(p). \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 pp-adic absolute value. Since addition and multiplication of integers are uniformly continuous with respect to the pp-adic absolute value, they extend uniquely to a uniformly continuous addition and multiplication on p\mathbb{Z}_p. Thus p\mathbb{Z}_p is a topological ring.

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

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

Properties

Topology

The ring of pp-adic integers has the following properties:

Relation to profinite completion of the integers

Example

The profinite completion of the integers is

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

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

^pprime p. \widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p \,.

(e.g. Lenstra, example 2.2)

Definition

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

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

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

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

References

Introductions and surveys include

  • Dennis Sullivan, pp. 9 of Localization, Periodicity and Galois Symmetry (The 1970 MIT notes) edited by Andrew Ranicki, K-Monographs in Mathematics, Dordrecht: Springer (pdf)

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

  • Hendrik Lenstra, Profinite groups (pdf)

Revised on June 24, 2014 19:27:02 by Urs Schreiber (89.204.130.177)