Profinite integer

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

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

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

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

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

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

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.^[2]

There is a canonical paring

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

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

See also

• Supernatural number

Notes

1. Milne, Ch. I Example A. 5.
2. http://math.stackexchange.com/questions/233136/questions-on-some-maps-involving-rings-of-finite-adeles-and-their-unit-groups
3. Connes–Consani, § 2.4.
4. K. Conrad, The character group of Q

References

• Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580.
• Milne, Class Field Theory

External links

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