Profinite integer

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

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

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

Concretely the profinite integers will be the set of maps ${\displaystyle \upsilon }$ such that ${\displaystyle \upsilon (n)\in \mathbb {Z} /n\mathbb {Z} }$ and ${\displaystyle m\ |\ n\implies \upsilon (m)\equiv \upsilon (n){\bmod {m}}}$. Pointwise addition and multiplication makes it a (non-integral) commutative ring. If a sequence of integers converges modulo n for every n then the limit will exist as a profinite integer.

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

${\displaystyle \mathbb {Z} \hookrightarrow {\widehat {\mathbb {Z} }},\,n\mapsto (n{\bmod {1}},n{\bmod {2}},\dots ).}$

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 Pontryagin dual of ${\displaystyle \mathbb {Q} /\mathbb {Z} }$.