# Profinite integer

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

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

Concretely the profinite integers will be the set of maps $\upsilon$ such that $\upsilon (n)\in \mathbb {Z} /n\mathbb {Z}$ and $m\ |\ n\implies \upsilon (m)\equiv \upsilon (n){\bmod {m}}$ . 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 ${\overline {\mathbf {F} }}_{q}$ be the algebraic closure of a finite field $\mathbf {F} _{q}$ of order q. Then $\operatorname {Gal} ({\overline {\mathbf {F} }}_{q}/\mathbf {F} _{q})={\widehat {\mathbb {Z} }}$ .

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

$\mathbb {Z} \hookrightarrow {\widehat {\mathbb {Z} }},\,n\mapsto (n{\bmod {1}},n{\bmod {2}},\dots ).$ The tensor product ${\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q}$ is the ring of finite adeles $\mathbf {A} _{\mathbb {Q} ,f}=\prod _{p}{}^{'}\mathbb {Q} _{p}$ of $\mathbb {Q}$ where the prime ' means restricted product.

There is a canonical pairing

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

## See also

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