Profinite integer

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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

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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

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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

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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

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Topological properties

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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

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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

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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

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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

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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

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Notes

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  1. ^ a b Lenstra, Hendrik. "Profinite number theory" (PDF). Mathematical Association of America. Retrieved 11 August 2022.
  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". www.math.columbia.edu. Retrieved 2020-09-25.

References

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