In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) where the inverse limit of the quotient rings
By definition, this ring is the profinite completion of the integers
can also be understood as the direct product of rings where the index
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.
Pointwise addition and multiplication make it a commutative ring.
, there exists a unique continuous group homomorphism
has a unique representation in the factorial number system as
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
determine the value of the profinite integer mod
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.
Another way to understand the construction of the profinite integers is by using the Chinese remainder theorem.
of non-repeating primes, there is a ring isomorphism
will just be a map on the underlying decompositions where there are induced surjections
It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism
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
which is compact with its product topology by Tychonoff's theorem.
Note the topology on each finite group
The Pontryagin dual is explicitly constructed by the function[2]
That is, an element is a sequence that is integral except at a finite number of places.
of order q, the Galois group can be computed explicitly.
where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of
which gives a computation of the absolute Galois group of a finite field.
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 étale fundamental group of the algebraic torus
as well from the fundamental exact sequence in étale homotopy theory.
There is an analogous statement for local class field theory since every finite abelian extension of
is induced from a finite field extension