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

• p-adic number
• Supernatural number

References

• Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580.
• Milne, J.S. (2013-03-23). "Class Field Theory" (PDF). Archived from the original (PDF) on 2013-06-19. Retrieved 2020-06-07.

External links

• http://ncatlab.org/nlab/show/profinite+completion+of+the+integers
• https://web.archive.org/web/20150401092904/http://www.noncommutative.org/supernatural-numbers-and-adeles/