The profinite integers can be constructed as the set of sequences of residues represented as
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:
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 .
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
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.
The set of profinite integers has an induced topology in which it is a compactHausdorff space, coming from the fact that it can be seen as a closed subset of the infinite direct product
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[2]
where is the character of the adele (introduced below) induced by .[3]
of 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
Applications in Galois theory and Etale homotopy theoryedit
For the algebraic closure 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[5]
Relation with Etale fundamental groups of algebraic toriedit
This construction can be re-interpreted in many ways. One of them is from Etale homotopy theory which defines the Etale fundamental group as the profinite completion of automorphisms
where is an Etale cover. Then, the profinite integers are isomorphic to the group
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
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.
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[6]
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 .