Profinite integer

Short description: Number theory concept

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

ℤ^=lim<-ℤ/nℤ,

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

ℤ^=∏pℤp,

where the index p runs over all prime numbers, and ℤp 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 étale homotopy theory
* 2.3.1 Relation with étale 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 ℤ^ can be constructed as the set of sequences υ of residues represented as υ=(υ1mod1,υ2mod2,υ3mod3,…) such that m | n⟹υm≡υnmodm.

Pointwise addition and multiplication make it a commutative ring.

The ring of integers embeds into the ring of profinite integers by the canonical injection: η:ℤ↪ℤ^ where n↦(nmod1,nmod2,…). It is canonical since it satisfies the universal property of profinite groups that, given any profinite group H and any group homomorphism f:ℤ->H, there exists a unique continuous group homomorphism g:ℤ^->H with f=gη.

### Using Factorial number system

Every integer n≥0 has a unique representation in the factorial number system as n=∑i=1∞cii!with ci∈ℤ where 0≤ci≤i for every i, and only finitely many of c1,c2,c3,… are nonzero.

Its factorial number representation can be written as (⋯c3c2c1)!.

In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string (⋯c3c2c1)!, where each ci is an integer satisfying 0≤ci≤i.[1]

The digits c1,c2,c3,…,ck−1 determine the value of the profinite integer mod k!. More specifically, there is a ring homomorphism ℤ^->ℤ/k!ℤ sending (⋯c3c2c1)!↦∑i=1k−1cii!modk! 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=p1a1⋯pkak of non-repeating primes, there is a ring isomorphism ℤ/n≅ℤ/p1a1×⋯×ℤ/pkak from the theorem. Moreover, any surjection ℤ/n->ℤ/m will just be a map on the underlying decompositions where there are induced surjections ℤ/piai->ℤ/pibi since we must have ai≥bi. It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism ℤ^≅∏pℤp with the direct product of p-adic integers.

Explicitly, the isomorphism is ϕ:∏pℤp->ℤ^ by ϕ((n2,n3,n5,⋯))(k)=∏qnqmodk where q ranges over all prime-power factors pidi of k, that is, k=∏i=1lpidi for some different prime numbers p1,...,pl.

## 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 ℤ^⊂∏n=1∞ℤ/nℤ which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group ℤ/nℤ is given as the discrete topology.

The topology on ℤ^ can be defined by the metric,[1] d(x,y)=1min⁡{k∈ℤ>0:x≢ymod(k+1)!}

Since addition of profinite integers is continuous, ℤ^ is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group.

In fact, the Pontryagin dual of ℤ^ is the abelian group ℚ/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[2] ℚ/ℤ×ℤ^->U(1),(q,a)↦χ(qa) where χ is the character of the adele (introduced below) 𝐀ℚ,f induced by ℚ/ℤ->U(1),α↦e2πiα.[3]

### Relation with adeles

The tensor product ℤ^⊗ℤℚ is the ring of finite adeles 𝐀ℚ,f=∏p′ℚp of ℚ where the symbol  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 𝐀ℚ≅ℝ×(ℤ^⊗ℤℚ)

### Applications in Galois theory and étale homotopy theory

For the algebraic closure 𝐅‾q of a finite field 𝐅q of order q, the Galois group can be computed explicitly. From the fact Gal(𝐅qn/𝐅q)≅ℤ/nℤ where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of 𝐅q is given by the inverse limit of the groups ℤ/nℤ, so its Galois group is isomorphic to the group of profinite integers[5] Gal⁡(𝐅‾q/𝐅q)≅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 π1et(X) as the profinite completion of automorphisms π1et(X)=limi∈IAut(Xi/X) where Xi->X is an étale cover. Then, the profinite integers are isomorphic to the group π1et(Spec(𝐅q))≅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 ℤ^↪π1et(𝔾m) since the covering maps come from the polynomial maps (⋅)n:𝔾m->𝔾m from the map of commutative rings f:ℤ[x,x−1]->ℤ[x,x−1] sending x↦xn since 𝔾m=Spec(ℤ[x,x−1]). If the algebraic torus is considered over a field k, then the étale fundamental group π1et(𝔾m/Spec(k)) contains an action of Gal(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 ℚ, the abelianization of its absolute Galois group Gal(ℚ‾/ℚ)ab is intimately related to the associated ring of adeles 𝔸ℚ and the group of profinite integers. In particular, there is a map, called the Artin map[6] Ψℚ:𝔸ℚ×/ℚ×->Gal(ℚ‾/ℚ)ab which is an isomorphism. This quotient can be determined explicitly as

𝔸ℚ×/ℚ×≅(ℝ×ℤ^)/ℤ=lim<-(ℝ/mℤ)=limx↦xmS1=ℤ^

giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of K/ℚp is induced from a finite field extension 𝔽pn/𝔽p.

## See also

* 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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