Profinite integer

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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

Notes

References

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  • Milne, Class Field Theory

External links

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