Profinite integer

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In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)

${\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} =\prod _{p}\mathbb {Z} _{p}}$

where the inverse limit

${\displaystyle \varprojlim \mathbb {Z} /n\mathbb {Z} }$

indicates the profinite completion of ${\displaystyle \mathbb {Z} }$, the index ${\displaystyle p}$ runs over all prime numbers, and ${\displaystyle \mathbb {Z} _{p}}$ is the ring of p-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic tractable example of a profinite group.