Profinite integer

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 product

${\displaystyle {\hat {\mathbb {Z} }}\subset \prod _{n}\mathbb {Z} /n\mathbb {Z} }$

which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ is given as the discrete topology. Since addition of profinite integers is continuous, ${\displaystyle {\widehat {\mathbb {Z} }}}$ is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group. In fact the Pontryagin dual of ${\displaystyle {\widehat {\mathbb {Z} }}}$ is the discrete abelian group ${\displaystyle \mathbb {Q} /\mathbb {Z} }$. This fact is exhibited by the pairing

${\displaystyle \mathbb {Q} /\mathbb {Z} \times {\widehat {\mathbb {Z} }}\to U(1),\,(q,a)\mapsto \chi (qa)}$[1]

where ${\displaystyle \chi }$ is the character of ${\displaystyle \mathbf {A} _{\mathbb {Q} ,f}}$ induced by ${\displaystyle \mathbb {Q} /\mathbb {Z} \to U(1),\,\alpha \mapsto e^{2\pi i\alpha }}$.[2]

The tensor product ${\displaystyle {\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} }$ is the ring of finite adeles

${\displaystyle \mathbf {A} _{\mathbb {Q} ,f}=\prod _{p}{}^{'}\mathbb {Q} _{p}}$

of ${\displaystyle \mathbb {Q} }$ where the symbol ${\displaystyle '}$ means restricted product.[3] There is an isomorphism

${\displaystyle \mathbf {A} _{\mathbb {Q} }\cong \mathbb {R} \times ({\hat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )}$

For the algebraic closure ${\displaystyle {\overline {\mathbf {F} }}_{q}}$ of a finite field ${\displaystyle \mathbf {F} _{q}}$ of order q, the Galois group can be computed explicitly. From the fact ${\displaystyle {\text{Gal}}(\mathbf {F} _{q^{n}}/\mathbf {F} _{q})\cong \mathbb {Z} /n\mathbb {Z} }$ where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of ${\displaystyle \mathbf {F} _{q}}$ is given by the inverse limit of the groups ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$, so its Galois group is isomorphic to the group of profinite integers[4]

${\displaystyle \operatorname {Gal} ({\overline {\mathbf {F} }}_{q}/\mathbf {F} _{q})\cong {\widehat {\mathbb {Z} }}}$

which gives a computation of the absolute Galois group of a finite field.

This construction can be re-interpreted in many ways. One of them is from Etale homotopy theory which defines the Etale fundamental group ${\displaystyle \pi _{1}^{et}(X)}$ as the profinite completion of automorphisms

${\displaystyle \pi _{1}^{et}(X)=\lim _{i\in I}{\text{Aut}}(X_{i}/X)}$

where ${\displaystyle X_{i}\to X}$ is an Etale cover. Then, the profinite integers are isomorphic to the group

${\displaystyle \pi _{1}^{et}({\text{Spec}}(\mathbf {F} _{q}))\cong {\hat {\mathbb {Z} }}}$

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

${\displaystyle {\hat {\mathbb {Z} }}\hookrightarrow \pi _{1}^{et}(\mathbb {G} _{m})}$

since the covering maps come from the polynomial maps

${\displaystyle (\cdot )^{n}:\mathbb {G} _{m}\to \mathbb {G} _{m}}$

from the map of commutative rings

${\displaystyle f:\mathbb {Z} [x,x^{-1}]\to \mathbb {Z} [x,x^{-1}]}$ sending ${\displaystyle x\mapsto x^{n}}$

since ${\displaystyle \mathbb {G} _{m}={\text{Spec}}(\mathbb {Z} [x,x^{-1}])}$. If the algebraic torus is considered over a field ${\displaystyle k}$, then the Etale fundamental group ${\displaystyle \pi _{1}^{et}(\mathbb {G} _{m}/{\text{Spec(k)}})}$ contains an action of ${\displaystyle {\text{Gal}}({\overline {k}}/k)}$ as well from the fundamental exact sequence in etale homotopy theory.

Class field theory is a branch of algebraic number theory studying the abelian field extensions of a field. Given the global field ${\displaystyle \mathbb {Q} }$, the abelianization of its absolute Galois group

${\displaystyle {\text{Gal}}({\overline {\mathbb {Q} }}/\mathbb {Q} )^{ab}}$

is intimately related to the associated ring of adeles ${\displaystyle \mathbb {A} _{\mathbb {Q} }}$ and the group of profinite integers. In particular, there is a map, called the Artin map[5]

${\displaystyle \Psi _{\mathbb {Q} }:\mathbb {A} _{\mathbb {Q} }^{\times }/\mathbb {Q} ^{\times }\to {\text{Gal}}({\overline {\mathbb {Q} }}/\mathbb {Q} )^{ab}}$

which is an isomorphism. This quotient can be determined explicitly as

${\displaystyle {\begin{aligned}\mathbb {A} _{\mathbb {Q} }^{\times }/\mathbb {Q} ^{\times }&\cong (\mathbb {R} \times {\hat {\mathbb {Z} }})/\mathbb {Z} \\&={\underset {\leftarrow }{\lim }}\mathbb {(} {\mathbb {R} }/m\mathbb {Z} )\\&={\underset {x\mapsto x^{m}}{\lim }}S^{1}\\&={\hat {\mathbb {Z} }}\end{aligned}}}$

giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of ${\displaystyle K/\mathbb {Q} _{p}}$ is induced from a finite field extension ${\displaystyle \mathbb {F} _{p^{n}}/\mathbb {F} _{p}}$.