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 Profinite integer ： ウィキペディア英語版
Profinite integer
In mathematics, a profinite integer is an element of the ring
:$\widehat_p$
where ''p'' runs over all prime numbers, $\mathbb_p$ is the ring of ''p''-adic integers and $\widehat/n$ (profinite completion).
Example: Let $\overline_q$ of order ''q''. Then $\operatorname\left(\overline_q\right) = \widehat \hookrightarrow \widehat\right\} \otimes_$ is the ring of finite adeles $\mathbf_ = \prod_p_p$ of $\mathbb$ where the prime ' means restricted product.〔http://math.stackexchange.com/questions/233136/questions-on-some-maps-involving-rings-of-finite-adeles-and-their-unit-groups〕
There is a canonical paring
:$\mathbb/\mathbb \times \widehat_$ induced by $\mathbb/\mathbb \to U\left(1\right), \, \alpha \mapsto e^$.〔K. Conrad, (The character group of Q )〕 The pairing identifies $\widehat/\mathbb$.