Profinite integer

In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)

: $\widehat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z},$

where the inverse limit of the quotient rings $\mathbb{Z}/n\mathbb{Z}$ runs through all natural numbers $n$ , partially ordered by divisibility. By definition, this ring is the profinite completion of the integers $\mathbb{Z}$ . By the Chinese remainder theorem, $\widehat{\mathbb{Z}}$ can also be understood as the direct product of rings

: $\widehat{\mathbb{Z}} = \prod_p \mathbb{Z}_p,$

where the index $p$ runs over all prime numbers, and $\mathbb{Z}_p$ is the ring of p-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic tractable example of a profinite group.

Construction

The profinite integers $\widehat{\Z}$ can be constructed as the set of sequences $\upsilon$ of residues represented as

$\upsilon = (\upsilon_1 \bmod 1, ~ \upsilon_2 \bmod 2, ~ \upsilon_3 \bmod 3, ~ \ldots)$

such that $m \ |\ n \implies \upsilon_m \equiv \upsilon_n \bmod m$ .

Pointwise addition and multiplication make it a commutative ring.

The ring of integers embeds into the ring of profinite integers by the canonical injection:

$\eta: \mathbb{Z} \hookrightarrow \widehat{\mathbb{Z}}$ where $n \mapsto (n \bmod 1, n \bmod 2, \dots).$

It is canonical since it satisfies the universal property of profinite groups that, given any profinite group $H$ and any group homomorphism $f : \Z \rightarrow H$ , there exists a unique continuous group homomorphism $g : \widehat{\Z} \rightarrow H$ with $f = g \eta$ .

= Using Factorial number system =

Every integer $n \ge 0$ has a unique representation in the factorial number system as

$n = \sum_{i=1}^\infty c_i i! \qquad \text{with } c_i \in \Z$

where $0 \le c_i \le i$ for every $i$ , and only finitely many of $c_1,c_2,c_3,\ldots$ are nonzero.

Its factorial number representation can be written as $(\cdots c_3 c_2 c_1)_!$ .

In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string $(\cdots c_3 c_2 c_1)_!$ , where each $c_i$ is an integer satisfying $0 \le c_i \le i$ .{{cite web |last1=Lenstra |first1=Hendrik |title=Profinite number theory |url=https://www.maa.org/sites/default/files/images/mathfest/2016/pntt.pdf |website=Mathematical Association of America |access-date=11 August 2022}}

The digits $c_1, c_2, c_3, \ldots, c_{k-1}$ determine the value of the profinite integer mod $k!$ . More specifically, there is a ring homomorphism $\widehat{\Z}\to \Z / k! \, \Z$ sending

$(\cdots c_3 c_2 c_1)_! \mapsto \sum_{i=1}^{k-1} c_i i! \mod k!$

The difference of a profinite integer from an integer is that the "finitely many nonzero digits" condition is dropped, allowing for its factorial number representation to have infinitely many nonzero digits.

= Using the Chinese Remainder theorem =

Another way to understand the construction of the profinite integers is by using the Chinese remainder theorem. Recall that for an integer $n$ with prime factorization

$n = p_1^{a_1}\cdots p_k^{a_k}$

of non-repeating primes, there is a ring isomorphism

$\mathbb{Z}/n \cong \mathbb{Z}/p_1^{a_1}\times \cdots \times \mathbb{Z}/p_k^{a_k}$

from the theorem. Moreover, any surjection

$\mathbb{Z}/n \to \mathbb{Z}/m$

will just be a map on the underlying decompositions where there are induced surjections

$\mathbb{Z}/p_i^{a_i} \to \mathbb{Z}/p_i^{b_i}$

since we must have $a_i \geq b_i$ . It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism

$\widehat{\mathbb{Z}} \cong \prod_p \mathbb{Z}_p$

with the direct product of p-adic integers.

Explicitly, the isomorphism is $\phi: \prod_p \mathbb{Z}_p \to \widehat\Z$ by

$\phi((n_2, n_3, n_5, \cdots))(k) = \prod_{q} n_q \mod k$

where $q$ ranges over all prime-power factors $p_i^{d_i}$ of $k$ , that is, $k = \prod_{i=1}^l p_i^{d_i}$ for some different prime numbers $p_1, ..., p_l$ .

Relations

= Topological properties =

The set of profinite integers has an induced topology in which it is a compact Hausdorff space, coming from the fact that it can be seen as a closed subset of the infinite direct product

$\widehat{\mathbb{Z}} \subset \prod_{n=1}^\infty \mathbb{Z}/n\mathbb{Z}$

which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group $\mathbb{Z}/n\mathbb{Z}$ is given as the discrete topology.

The topology on $\widehat{\Z}$ can be defined by the metric,

$d(x,y) = \frac1{ \min\{ k \in \Z_{>0} : x \not\equiv y \bmod{(k+1)!} \} }$

Since addition of profinite integers is continuous, $\widehat{\mathbb{Z}}$ is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group.

In fact, the Pontryagin dual of $\widehat{\mathbb{Z}}$ is the abelian group $\mathbb{Q}/\mathbb{Z}$ equipped with the discrete topology (note that it is not the subset topology inherited from $\R/\Z$ , which is not discrete). The Pontryagin dual is explicitly constructed by the function{{harvnb|Connes|Consani|2015|loc=§ 2.4.}}

$\mathbb{Q}/\mathbb{Z} \times \widehat{\mathbb{Z}} \to U(1), \, (q, a) \mapsto \chi(qa)$

where $\chi$ is the character of the adele (introduced below) $\mathbf{A}_{\mathbb{Q}, f}$ induced by $\mathbb{Q}/\mathbb{Z} \to U(1), \, \alpha \mapsto e^{2\pi i\alpha}$ .K. Conrad, [http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/characterQ.pdf The character group of Q]

= Relation with adeles =

The tensor product $\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}$ is the ring of finite adeles

$\mathbf{A}_{\mathbb{Q}, f} = {\prod_p}' \mathbb{Q}_p$

of $\mathbb{Q}$ where the symbol $'$ means restricted product. That is, an element is a sequence that is integral except at a finite number of places.[https://math.stackexchange.com/q/233136 Questions on some maps involving rings of finite adeles and their unit groups]. There is an isomorphism

$\mathbf{A}_\mathbb{Q} \cong \mathbb{R}\times(\hat{\mathbb{Z}}\otimes_\mathbb{Z}\mathbb{Q})$

= Applications in Galois theory and étale homotopy theory =

For the algebraic closure $\overline{\mathbf{F}}_q$ of a finite field $\mathbf{F}_q$ of order q, the Galois group can be computed explicitly. From the fact $\text{Gal}(\mathbf{F}_{q^n}/\mathbf{F}_q) \cong \mathbb{Z}/n\mathbb{Z}$ where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of $\mathbf{F}_q$ is given by the inverse limit of the groups $\mathbb{Z}/n\mathbb{Z}$ , so its Galois group is isomorphic to the group of profinite integers{{harvnb|Milne|2013|loc=Ch. I Example A. 5.}}

$\operatorname{Gal}(\overline{\mathbf{F}}_q/\mathbf{F}_q) \cong \widehat{\mathbb{Z}}$

which gives a computation of the absolute Galois group of a finite field.

== Relation with étale fundamental groups of algebraic tori ==

This construction can be re-interpreted in many ways. One of them is from étale homotopy type which defines the étale fundamental group $\pi_1^{et}(X)$ as the profinite completion of automorphisms

$\pi_1^{et}(X) = \lim_{i \in I} \text{Aut}(X_i/X)$

where $X_i \to X$ is an étale cover. Then, the profinite integers are isomorphic to the group

$\pi_1^{et}(\text{Spec}(\mathbf{F}_q)) \cong \hat{\mathbb{Z}}$

from the earlier computation of the profinite Galois group. In addition, there is an embedding of the profinite integers inside the étale fundamental group of the algebraic torus

$\hat{\mathbb{Z}} \hookrightarrow \pi_1^{et}(\mathbb{G}_m)$

since the covering maps come from the polynomial maps

$(\cdot)^n:\mathbb{G}_m \to \mathbb{G}_m$

from the map of commutative rings

$f:\mathbb{Z}[x,x^{-1}] \to \mathbb{Z}[x,x^{-1}]$ sending $x \mapsto x^n$

since $\mathbb{G}_m = \text{Spec}(\mathbb{Z}[x,x^{-1}])$ . If the algebraic torus is considered over a field $k$ , then the étale fundamental group $\pi_1^{et}(\mathbb{G}_m/\text{Spec(k)})$ contains an action of $\text{Gal}(\overline{k}/k)$ as well from the fundamental exact sequence in étale homotopy theory.

= Class field theory and the profinite integers =

Class field theory is a branch of algebraic number theory studying the abelian field extensions of a field. Given the global field $\mathbb{Q}$ , the abelianization of its absolute Galois group

$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^{ab}$

is intimately related to the associated ring of adeles $\mathbb{A}_\mathbb{Q}$ and the group of profinite integers. In particular, there is a map, called the Artin map{{Cite web|title=Class field theory - lccs|url=http://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html#sec13|access-date=2020-09-25|website=www.math.columbia.edu}}

$\Psi_\mathbb{Q}:\mathbb{A}_\mathbb{Q}^\times / \mathbb{Q}^\times \to
\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^{ab}$

which is an isomorphism. This quotient can be determined explicitly as

$\begin{align}
\mathbb{A}_\mathbb{Q}^\times/\mathbb{Q}^\times &\cong (\mathbb{R}\times \hat{\mathbb{Z}})/\mathbb{Z} \\
&= \underset{\leftarrow}{\lim} \mathbb({\mathbb{R}}/m\mathbb{Z}) \\
&= \underset{x \mapsto x^m}{\lim} S^1 \\
&= \hat{\mathbb{Z}}
\end{align}$

giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of $K/\mathbb{Q}_p$ is induced from a finite field extension $\mathbb{F}_{p^n}/\mathbb{F}_p$ .

Notes

References

{{cite arXiv |first1=Alain |last1=Connes |first2=Caterina |last2=Consani|author2-link=Caterina Consani |eprint=1502.05580 |title=Geometry of the arithmetic site |date=2015 |class=math.AG }}
{{cite web|url=http://www.jmilne.org/math/CourseNotes/CFT.pdf |title=Class Field Theory |last=Milne |first=J.S. |date=2013-03-23 |access-date=2020-06-07 |archive-url=https://web.archive.org/web/20130619104611/http://www.jmilne.org/math/CourseNotes/CFT.pdf |archive-date=2013-06-19}}

External links

http://ncatlab.org/nlab/show/profinite+completion+of+the+integers
https://web.archive.org/web/20150401092904/http://www.noncommutative.org/supernatural-numbers-and-adeles/
https://euro-math-soc.eu/system/files/news/Hendrik%20Lenstra_Profinite%20number%20theory.pdf

Category:Algebraic number theory

Category:P-adic numbers

Category:Ring theory