p-adic integer



Formal geometry

Arithmetic geometry



For each prime number pp the ring of pp-adic integers p\mathbb{Z}_p is the formal completion of the ring \mathbb{Z} at the prime ideal (p)(p). Geometrically this means that p\mathbb{Z}_p is the ring of functions on a formal neighbourhood of pp inside Spec(Z) (this is discussed in more detail below). Algebraically it means that the elements in p\mathbb{Z}_p look like formal power series where the formal variable is the prime number pp.


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).

There is also this characterization:


There is a short exact sequence

0 (p)p() (p)/p0. 0 \to \mathbb{Z}_{(p)} \overset{p \cdot (-)}{\longrightarrow} \mathbb{Z}_{(p)} \longrightarrow \mathbb{Z}/p\mathbb{Z} \to 0 \,.

Consider the following commuting diagram

/p 3 p() /p 4 /p /p 2 p() /p 3 /p /p p() /p 2 /p 0 /p /p. \array{ \vdots && \vdots && \vdots \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p^3\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^4 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p^2\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^3 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{Z}/p\mathbb{Z} &\overset{p\cdot (-)}{\longrightarrow}& \mathbb{Z}/p^2 \mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} \\ \downarrow && \downarrow && \downarrow \\ 0 &\longrightarrow& \mathbb{Z}/p\mathbb{Z} &\longrightarrow& \mathbb{Z}/p\mathbb{Z} } \,.

Each horizontal sequence is exact. Taking the limit over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.



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

Relation to profinite completion of the integers


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)


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.

Pontrajgin duality to Prüfer pp-group

Under Pontryagin duality, the abelian group underlying p\mathbb{Z}_p maps to the Prüfer p-group [p 1]/\mathbb{Z}[p^{-1}]/\mathbb{Z}, see at Pontryagin duality for torsion abelian groups.

[p 1]/ / / hom(,/) p ^ \array{ &\mathbb{Z}[p^{-1}]/\mathbb{Z} &\hookrightarrow& \mathbb{Q}/\mathbb{Z} &\hookrightarrow& \mathbb{R}/\mathbb{Z} \\ {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow \\ &\mathbb{Z}_p &\leftarrow& \hat \mathbb{Z} &\leftarrow& \mathbb{Z} }

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).

This plays a central role for instance in the function field analogy. It is highlighted for instance in (Hartl 06, 1.1, Buium 13, section 1.1.3). See also at arithmetic jet space and at ring of Witt vectors.

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization


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)

The synthetic differential geometry-aspect of the pp-adic numbers is highlighted for instance in

Revised on July 14, 2016 14:15:33 by Urs Schreiber (