Profinite integer explained

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

} = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_pwhere the inverse limit indicates the profinite completion of , the index runs over all prime numbers, and 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

can be constructed as the set of sequences of residues represented as$$\backslash upsilon\; =\; (\backslash upsilon\_1\; \backslash bmod\; 1,\; ~\; \backslash upsilon\_2\; \backslash bmod\; 2,\; ~\; \backslash upsilon\_3\; \backslash bmod\; 3,\; ~\; \backslash ldots)$$such that .

Pointwise addition and multiplication make it a commutative ring.

The ring of integers embeds into the ring of profinite integers by the canonical injection:$$\backslash eta:\; \backslash mathbb\; \backslash hookrightarrow\; \backslash widehat$$ where

It is canonical since it satisfies the universal property of profinite groups that, given any profinite group and any group homomorphism , there exists a unique continuous group homomorphism with .

Using Factorial number system

Every integer

has a unique representation in the factorial number system as$$n\; =\; \backslash sum\_^\backslash infty\; c\_i\; i!\; \backslash qquad\; \backslash text\; c\_i\; \backslash in\; \backslash Z$$where for every , and only finitely many of are nonzero.

Its factorial number representation can be written as

.

In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string

, where each is an integer satisfying .^[1]

The digits

determine the value of the profinite integer mod . More specifically, there is a ring homomorphism sending$$(\backslash cdots\; c\_3\; c\_2\; c\_1)\_!\; \backslash mapsto\; \backslash sum\_^\; c\_i\; i!\; \backslash mod\; k!$$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

with prime factorization$$n\; =\; p\_1^\backslash cdots\; p\_k^$$of non-repeating primes, there is a ring isomorphism$$\backslash mathbb/n\; \backslash cong\; \backslash mathbb/p\_1^\backslash times\; \backslash cdots\; \backslash times\; \backslash mathbb/p\_k^$$from the theorem. Moreover, any surjection$$\backslash mathbb/n\; \backslash to\; \backslash mathbb/m$$will just be a map on the underlying decompositions where there are induced surjections$$\backslash mathbb/p\_i^\; \backslash to\; \backslash mathbb/p\_i^$$since we must have . It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism$$\backslash widehat\; \backslash cong\; \backslash prod\_p\; \backslash mathbb\_p$$with the direct product of p-adic integers.

Explicitly, the isomorphism is

by$$\backslash phi((n\_2,\; n\_3,\; n\_5,\; \backslash cdots))(k)\; =\; \backslash prod\_\; n\_q\; \backslash mod\; k$$where ranges over all prime-power factors of , that is, for some different prime numbers .

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$$\backslash widehat\; \backslash subset\; \backslash prod\_^\backslash infty\; \backslash mathbb/n\backslash mathbb$$which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group

is given as the discrete topology.

The topology on

can be defined by the metric,$$d(x,y)\; =\; \backslash frac1$$

Since addition of profinite integers is continuous,

} is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group.

In fact, the Pontryagin dual of

} is the abelian group equipped with the discrete topology (note that it is not the subset topology inherited from , which is not discrete). The Pontryagin dual is explicitly constructed by the function$$\backslash mathbb/\backslash mathbb\; \backslash times\; \backslash widehat\; \backslash to\; U(1),\; \backslash ,\; (q,\; a)\; \backslash mapsto\; \backslash chi(qa)$$where is the character of the adele (introduced below) induced by .^[2]

Relation with adeles

The tensor product

} \otimes_ \mathbb is the ring of finite adeles$$\backslash mathbf\_\; =\; \text{'}\; \backslash mathbb\_p$$of where the symbol means restricted product. That is, an element is a sequence that is integral except at a finite number of places.^[3] There is an isomorphism$$\backslash mathbf\_\backslash mathbb\; \backslash cong\; \backslash mathbb\backslash times(\backslash hat\backslash otimes\_\backslash mathbb\backslash mathbb)$$

Applications in Galois theory and Etale homotopy theory

}_q of a finite field of order q, the Galois group can be computed explicitly. From the fact where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of is given by the inverse limit of the groups , so its Galois group is isomorphic to the group of profinite integers $$\backslash operatorname(\backslash overline\_q/\backslash mathbf\_q)\; \backslash cong\; \backslash widehat$$which gives a computation of the absolute Galois group of a finite field.

Relation with Etale fundamental groups of algebraic tori

as the profinite completion of automorphisms$$\backslash pi\_1^(X)\; =\; \backslash lim\_\; \backslash text(X\_i/X)$$where is an Etale cover. Then, the profinite integers are isomorphic to the group$$\backslash pi\_1^(\backslash text(\backslash mathbf\_q))\; \backslash cong\; \backslash hat$$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$$\backslash hat\; \backslash hookrightarrow\; \backslash pi\_1^(\backslash mathbb\_m)$$since the covering maps come from the polynomial maps$$(\backslash cdot)^n:\backslash mathbb\_m\; \backslash to\; \backslash mathbb\_m$$from the map of commutative rings$$f:\backslash mathbb[x,x^\{-1\}]\; \backslash to\; \backslash mathbb[x,x^\{-1\}]$$ sending since . If the algebraic torus is considered over a field , then the Etale fundamental group contains an action of as well from the fundamental exact sequence in etale homotopy theory.

Class field theory and the profinite integers

, the abelianization of its absolute Galois group$$\backslash text(\backslash overline/\backslash mathbb)^$$is intimately related to the associated ring of adeles and the group of profinite integers. In particular, there is a map, called the Artin map^[4] $$\backslash Psi\_\backslash mathbb:\backslash mathbb\_\backslash mathbb^\backslash times\; /\; \backslash mathbb^\backslash times\; \backslash to\backslash text(\backslash overline/\backslash mathbb)^$$which is an isomorphism. This quotient can be determined explicitly as

giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of

is induced from a finite field extension .

See also

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

References

• Alain . Connes . Caterina . Consani. Caterina Consani . 1502.05580 . Geometry of the arithmetic site . 2015 . math.AG .
• Web site: Class Field Theory . Milne . J.S. . 2013-03-23 . 2020-06-07 . https://web.archive.org/web/20130619104611/http://www.jmilne.org/math/CourseNotes/CFT.pdf . 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

Notes and References

1. Web site: Lenstra . Hendrik . Profinite number theory . Mathematical Association of America . 11 August 2022.
2. K. Conrad, The character group of Q
3. https://math.stackexchange.com/q/233136 Questions on some maps involving rings of finite adeles and their unit groups
4. Web site: Class field theory - lccs. 2020-09-25. www.math.columbia.edu.