# Profinite integer

(Redirected from Profinite integers)

In mathematics, a profinite integer is an element of the ring

${\displaystyle {\widehat {\mathbb {Z} }}=\prod _{p}\mathbb {Z} _{p}}$

where p runs over all prime numbers, ${\displaystyle \mathbb {Z} _{p}}$ is the ring of p-adic integers and ${\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} }$ (profinite completion).

Example: Let ${\displaystyle {\overline {\mathbf {F} }}_{q}}$ be the algebraic closure of a finite field ${\displaystyle \mathbf {F} _{q}}$ of order q. Then ${\displaystyle \operatorname {Gal} ({\overline {\mathbf {F} }}_{q}/\mathbf {F} _{q})={\widehat {\mathbb {Z} }}}$.[1]

A usual (rational) integer is a profinite integer since there is the canonical injection

${\displaystyle \mathbb {Z} \hookrightarrow {\widehat {\mathbb {Z} }},\,n\mapsto (n,n,\dots ).}$

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 prime ' means restricted product.[2]

There is a canonical pairing

${\displaystyle \mathbb {Q} /\mathbb {Z} \times {\widehat {\mathbb {Z} }}\to U(1),\,(q,a)\mapsto \chi (qa)}$[3]

where ${\displaystyle \chi }$ is the character of ${\displaystyle \mathbf {A} _{\mathbb {Q} ,f}}$ induced by ${\displaystyle \mathbb {Q} /\mathbb {Z} \to U(1),\,\alpha \mapsto e^{2\pi i\alpha }}$.[4] The pairing identifies ${\displaystyle {\widehat {\mathbb {Z} }}}$ with the Pontrjagin dual of ${\displaystyle \mathbb {Q} /\mathbb {Z} }$.

## Notes

1. ^ Milne, Ch. I Example A. 5.