Profinite integer

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

[math]\displaystyle{ \widehat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z} = \prod_p \mathbb{Z}_p }[/math]

where the inverse limit

[math]\displaystyle{ \varprojlim \mathbb{Z}/n\mathbb{Z} }[/math]

indicates the profinite completion of [math]\displaystyle{ \mathbb{Z} }[/math], the index [math]\displaystyle{ p }[/math] runs over all prime numbers, and [math]\displaystyle{ \mathbb{Z}_p }[/math] 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.

## Contents

* 1 Construction
* 1.1 Using Factorial number system
* 1.2 Using the Chinese Remainder theorem
* 2 Relations
* 2.1 Topological properties
* 2.2 Relation with adeles
* 2.3 Applications in Galois theory and Etale homotopy theory
* 2.3.1 Relation with Etale fundamental groups of algebraic tori
* 2.4 Class field theory and the profinite integers
* 3 See also
* 4 Notes
* 5 References
* 6 External links

## Construction

The profinite integers [math]\displaystyle{ \widehat{\Z} }[/math] can be constructed as the set of sequences [math]\displaystyle{ \upsilon }[/math] of residues represented as [math]\displaystyle{ \upsilon = (\upsilon_1 \bmod 1, ~ \upsilon_2 \bmod 2, ~ \upsilon_3 \bmod 3, ~ \ldots) }[/math] such that [math]\displaystyle{ m \ |\ n \implies \upsilon_m \equiv \upsilon_n \bmod m }[/math].

Pointwise addition and multiplication make it a commutative ring.

The ring of integers embeds into the ring of profinite integers by the canonical injection: [math]\displaystyle{ \eta: \mathbb{Z} \hookrightarrow \widehat{\mathbb{Z}} }[/math] where [math]\displaystyle{ n \mapsto (n \bmod 1, n \bmod 2, \dots). }[/math] It is canonical since it satisfies the universal property of profinite groups that, given any profinite group [math]\displaystyle{ H }[/math] and any group homomorphism [math]\displaystyle{ f : \Z \rightarrow H }[/math], there exists a unique continuous group homomorphism [math]\displaystyle{ g : \widehat{\Z} \rightarrow H }[/math] with [math]\displaystyle{ f = g \eta }[/math].

### Using Factorial number system

Every integer [math]\displaystyle{ n \ge 0 }[/math] has a unique representation in the factorial number system as [math]\displaystyle{ n = \sum_{i=1}^\infty c_i i! \qquad \text{with } c_i \in \Z }[/math] where [math]\displaystyle{ 0 \le c_i \le i }[/math] for every [math]\displaystyle{ i }[/math], and only finitely many of [math]\displaystyle{ c_1,c_2,c_3,\ldots }[/math] are nonzero.

Its factorial number representation can be written as [math]\displaystyle{ (\cdots c_3 c_2 c_1)_! }[/math].

In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string [math]\displaystyle{ (\cdots c_3 c_2 c_1)_! }[/math], where each [math]\displaystyle{ c_i }[/math] is an integer satisfying [math]\displaystyle{ 0 \le c_i \le i }[/math].[1]

The digits [math]\displaystyle{ c_1, c_2, c_3, \ldots, c_{k-1} }[/math] determine the value of the profinite integer mod [math]\displaystyle{ k! }[/math]. More specifically, there is a ring homomorphism [math]\displaystyle{ \widehat{\Z}\to \Z / k! \, \Z }[/math] sending [math]\displaystyle{ (\cdots c_3 c_2 c_1)_! \mapsto \sum_{i=1}^{k-1} c_i i! \mod k! }[/math] 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 [math]\displaystyle{ n }[/math] with prime factorization [math]\displaystyle{ n = p_1^{a_1}\cdots p_k^{a_k} }[/math] of non-repeating primes, there is a ring isomorphism [math]\displaystyle{ \mathbb{Z}/n \cong \mathbb{Z}/p_1^{a_1}\times \cdots \times \mathbb{Z}/p_k^{a_k} }[/math] from the theorem. Moreover, any surjection [math]\displaystyle{ \mathbb{Z}/n \to \mathbb{Z}/m }[/math] will just be a map on the underlying decompositions where there are induced surjections [math]\displaystyle{ \mathbb{Z}/p_i^{a_i} \to \mathbb{Z}/p_i^{b_i} }[/math] since we must have [math]\displaystyle{ a_i \geq b_i }[/math]. It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism [math]\displaystyle{ \widehat{\mathbb{Z}} \cong \prod_p \mathbb{Z}_p }[/math] with the direct product of p-adic integers.

Explicitly, the isomorphism is [math]\displaystyle{ \phi: \prod_p \mathbb{Z}_p \to \widehat\Z }[/math] by [math]\displaystyle{ \phi((n_2, n_3, n_5, \cdots))(k) = \prod_{q} n_q \mod k }[/math] where [math]\displaystyle{ q }[/math] ranges over all prime-power factors [math]\displaystyle{ p_i^{d_i} }[/math] of [math]\displaystyle{ k }[/math], that is, [math]\displaystyle{ k = \prod_{i=1}^l p_i^{d_i} }[/math] for some different prime numbers [math]\displaystyle{ p_1, ..., p_l }[/math].

## 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 [math]\displaystyle{ \widehat{\mathbb{Z}} \subset \prod_{n=1}^\infty \mathbb{Z}/n\mathbb{Z} }[/math] which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group [math]\displaystyle{ \mathbb{Z}/n\mathbb{Z} }[/math] is given as the discrete topology.

The topology on [math]\displaystyle{ \widehat{\Z} }[/math] can be defined by the metric,[1] [math]\displaystyle{ d(x,y) = \frac1{ \min\\{ k \in \Z_{\gt 0} : x \not\equiv y \bmod{(k+1)!} \\} } }[/math]

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

In fact, the Pontryagin dual of [math]\displaystyle{ \widehat{\mathbb{Z}} }[/math] is the abelian group [math]\displaystyle{ \mathbb{Q}/\mathbb{Z} }[/math] equipped with the discrete topology (note that it is not the subset topology inherited from [math]\displaystyle{ \R/\Z }[/math], which is not discrete). The Pontryagin dual is explicitly constructed by the function[2] [math]\displaystyle{ \mathbb{Q}/\mathbb{Z} \times \widehat{\mathbb{Z}} \to U(1), \, (q, a) \mapsto \chi(qa) }[/math] where [math]\displaystyle{ \chi }[/math] is the character of the adele (introduced below) [math]\displaystyle{ \mathbf{A}_{\mathbb{Q}, f} }[/math] induced by [math]\displaystyle{ \mathbb{Q}/\mathbb{Z} \to U(1), \, \alpha \mapsto e^{2\pi i\alpha} }[/math].[3]

### Relation with adeles

The tensor product [math]\displaystyle{ \widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} }[/math] is the ring of finite adeles [math]\displaystyle{ \mathbf{A}_{\mathbb{Q}, f} = {\prod_p}' \mathbb{Q}_p }[/math] of [math]\displaystyle{ \mathbb{Q} }[/math] where the symbol [math]\displaystyle{ ' }[/math] means restricted product. That is, an element is a sequence that is integral except at a finite number of places.[4] There is an isomorphism [math]\displaystyle{ \mathbf{A}_\mathbb{Q} \cong \mathbb{R}\times(\hat{\mathbb{Z}}\otimes_\mathbb{Z}\mathbb{Q}) }[/math]

### Applications in Galois theory and Etale homotopy theory

For the algebraic closure [math]\displaystyle{ \overline{\mathbf{F}}_q }[/math] of a finite field [math]\displaystyle{ \mathbf{F}_q }[/math] of order q, the Galois group can be computed explicitly. From the fact [math]\displaystyle{ \text{Gal}(\mathbf{F}_{q^n}/\mathbf{F}_q) \cong \mathbb{Z}/n\mathbb{Z} }[/math] where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of [math]\displaystyle{ \mathbf{F}_q }[/math] is given by the inverse limit of the groups [math]\displaystyle{ \mathbb{Z}/n\mathbb{Z} }[/math], so its Galois group is isomorphic to the group of profinite integers[5] [math]\displaystyle{ \operatorname{Gal}(\overline{\mathbf{F}}_q/\mathbf{F}_q) \cong \widehat{\mathbb{Z}} }[/math] which gives a computation of the absolute Galois group of a finite field.

#### Relation with Etale fundamental groups of algebraic tori

This construction can be re-interpreted in many ways. One of them is from Etale homotopy theory which defines the Etale fundamental group [math]\displaystyle{ \pi_1^{et}(X) }[/math] as the profinite completion of automorphisms [math]\displaystyle{ \pi_1^{et}(X) = \lim_{i \in I} \text{Aut}(X_i/X) }[/math] where [math]\displaystyle{ X_i \to X }[/math] is an Etale cover. Then, the profinite integers are isomorphic to the group [math]\displaystyle{ \pi_1^{et}(\text{Spec}(\mathbf{F}_q)) \cong \hat{\mathbb{Z}} }[/math] from the earlier computation of the profinite Galois group. In addition, there is an embedding of the profinite integers inside the Etale fundamental group of the algebraic torus [math]\displaystyle{ \hat{\mathbb{Z}} \hookrightarrow \pi_1^{et}(\mathbb{G}_m) }[/math] since the covering maps come from the polynomial maps [math]\displaystyle{ (\cdot)^n:\mathbb{G}_m \to \mathbb{G}_m }[/math] from the map of commutative rings [math]\displaystyle{ f:\mathbb{Z}[x,x^{-1}] \to \mathbb{Z}[x,x^{-1}] }[/math] sending [math]\displaystyle{ x \mapsto x^n }[/math] since [math]\displaystyle{ \mathbb{G}_m = \text{Spec}(\mathbb{Z}[x,x^{-1}]) }[/math]. If the algebraic torus is considered over a field [math]\displaystyle{ k }[/math], then the Etale fundamental group [math]\displaystyle{ \pi_1^{et}(\mathbb{G}_m/\text{Spec(k)}) }[/math] contains an action of [math]\displaystyle{ \text{Gal}(\overline{k}/k) }[/math] as well from the fundamental exact sequence in etale 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 [math]\displaystyle{ \mathbb{Q} }[/math], the abelianization of its absolute Galois group [math]\displaystyle{ \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^{ab} }[/math] is intimately related to the associated ring of adeles [math]\displaystyle{ \mathbb{A}_\mathbb{Q} }[/math] and the group of profinite integers. In particular, there is a map, called the Artin map[6] [math]\displaystyle{ \Psi_\mathbb{Q}:\mathbb{A}_\mathbb{Q}^\times / \mathbb{Q}^\times \to \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})^{ab} }[/math] which is an isomorphism. This quotient can be determined explicitly as

[math]\displaystyle{ \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} }[/math]

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

## See also

* p-adic number
* Ring of adeles
* Supernatural number

## Notes

1. ↑ 1.0 1.1 Lenstra, Hendrik. "Profinite number theory". https://www.maa.org/sites/default/files/images/mathfest/2016/pntt.pdf.
2. ↑ Connes & Consani 2015, § 2.4.
3. ↑ K. Conrad, The character group of Q
4. ↑ Questions on some maps involving rings of finite adeles and their unit groups.
5. ↑ Milne 2013, Ch. I Example A. 5.
6. ↑ "Class field theory - lccs". http://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html#sec13.

## References

* Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580 [math.AG].
* Milne, J.S. (2013-03-23). "Class Field Theory". http://www.jmilne.org/math/CourseNotes/CFT.pdf.

## 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 Lenstra_Profinite number theory.pdf

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