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
The profinite integers can be constructed as the set of sequences of residues represented as
Pointwise addition and multiplication make it a commutative ring.
The ring of integers embeds into the ring of profinite integers by the canonical injection:
Using Factorial number system
Every integer 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 , 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
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
Explicitly, the isomorphism is by
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
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]
Relation with adeles
The tensor product is the ring of finite adeles
Applications in Galois theory and Etale homotopy theory
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 tori
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
Class field theory and the profinite integers
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
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
Notes
- ^ a b Lenstra, Hendrik. "Profinite number theory" (PDF). Mathematical Association of America. Retrieved 11 August 2022.
- ^ Connes & Consani 2015, § 2.4.
- ^ K. Conrad, The character group of Q
- ^ Questions on some maps involving rings of finite adeles and their unit groups.
- ^ Milne 2013, Ch. I Example A. 5.
- ^ "Class field theory - lccs". www.math.columbia.edu. Retrieved 2020-09-25.
References
- Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580 [math.AG].
- Milne, J.S. (2013-03-23). "Class Field Theory" (PDF). Archived from the original (PDF) on 2013-06-19. Retrieved 2020-06-07.