
Profinite integer

In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)formula_1whereformula_2indicates the profinite completion of formula_3, the index formula_4 runs over all prime numbers, and formula_5 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 and relations.
Concretely the profinite integers will be the set of sequences formula_6 such that formula_7 and formula_8. Pointwise addition and multiplication makes it a commutative ring. If a sequence of integers converges modulo "n" for every "n" then the limit will exist as a profinite integer. There is an embedding of the integers into the ring of profinite integers since there is the canonical injection
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 productformula_11which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group formula_12 is given as the discrete topology. Since addition of profinite integers is continuous, formula_13 is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group. In fact the Pontryagin dual of formula_13 is the discrete abelian group formula_15. This fact is exhibited by the pairingformula_16where formula_17 is the character of formula_18 induced by formula_19.
Relation with adeles.
The tensor product formula_20 is the ring of finite adelesformula_21of formula_22 where the symbol formula_23 means restricted product. There is an isomorphismformula_24
Applications in Galois theory and Etale homotopy theory.
For the algebraic closure formula_25 of a finite field formula_26 of order "q," the Galois group can be computed explicitly. From the fact formula_27 where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of formula_26 is given by the inverse limit of the groups formula_12, so its Galois group is isomorphic to the group of profinite integers formula_30 which gives a computation of the absolute Galois group of a finite field.
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 formula_31 as the profinite completion of automorphismsformula_32where formula_33 is an Etale cover. Then, the profinite integers are isomorphic to the groupformula_34from 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 torusformula_35since the covering maps come from the polynomial mapsformula_36from the map of commutative ringsformula_37 sending formula_38since formula_39. If the algebraic torus is considered over a field formula_40, then the Etale fundamental group formula_41 contains an action of formula_42 as well from the fundamental exact sequence in etale homotopy theory.
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 formula_22, the abelianization of its absolute Galois groupformula_44is intimately related to the associated ring of adeles formula_45 and the group of profinite integers. In particular, there is a map, called the Artin mapformula_46which is an isomorphism. This quotient can be determined explicitly asformula_47giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of formula_48 is induced from a finite field extension formula_49.