In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)
where 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[edit]
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
.
Using Factorial number system[edit]
Every integer has a unique representation in the factorial number system as
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.
Using the Chinese Remainder theorem[edit]
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
of non-repeating primes, there is a
ring isomorphism
from the theorem. Moreover, any
surjection
will just be a map on the underlying decompositions where there are induced surjections
since we must have
. It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism
with the direct product of
p-adic integers.
Explicitly, the isomorphism is by
where
ranges over all prime-power factors
of
, that is,
for some different prime numbers
.
Relations[edit]
Topological properties[edit]
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
is given as the
discrete topology.
The topology on can be defined by the metric,[1]
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]
Relation with adeles[edit]
The tensor product is the ring of finite adeles
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 theory[edit]
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]
which gives a computation of the
absolute Galois group of a finite field.
Relation with Etale fundamental groups of algebraic tori[edit]
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
since the covering maps come from the
polynomial maps
from the map of
commutative rings
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[edit]
Class field theory is a branch of algebraic number theory studying the abelian field extensions of a field. Given the global field , the abelianization of its absolute Galois group
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 .
See also[edit]
References[edit]
External links[edit]