Profinite integer

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

${\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} ,}$

where the inverse limit of the quotient rings ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ runs through all natural numbers ${\displaystyle n}$, partially ordered by divisibility. By definition, this ring is the profinite completion of the integers ${\displaystyle \mathbb {Z} }$. By the Chinese remainder theorem, ${\displaystyle {\widehat {\mathbb {Z} }}}$ can also be understood as the direct product of rings

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

where 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.