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 $\widehat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n$ (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/