In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)
Z ^ = lim ← Z / n Z , {\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} ,}
where the inverse limit of the quotient rings Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } runs through all natural numbers n {\displaystyle n} , partially ordered by divisibility. By definition, this ring is the profinite completion of the integers Z {\displaystyle \mathbb {Z} } . By the Chinese remainder theorem, Z ^ {\displaystyle {\widehat {\mathbb {Z} }}} can also be understood as the direct product of rings
Z ^ = ∏ p Z p , {\displaystyle {\widehat {\mathbb {Z} }}=\prod _{p}\mathbb {Z} _{p},}
where the index p {\displaystyle p} runs over all prime numbers, and Z p {\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.
Construction The profinite integers Z ^ {\displaystyle {\widehat {\mathbb {Z} }}} can be constructed as the set of sequences υ {\displaystyle \upsilon } of residues represented as υ = ( υ 1 mod 1 , υ 2 mod 2 , υ 3 mod 3 , … ) {\displaystyle \upsilon =(\upsilon _{1}{\bmod {1}},~\upsilon _{2}{\bmod {2}},~\upsilon _{3}{\bmod {3}},~\ldots )} such that m | n ⟹ υ m ≡ υ n ( mod m ) {\displaystyle m\ |\ n\implies \upsilon _{m}\equiv \upsilon _{n}\!\!\!\!\!{\pmod {m}}} . Pointwise addition and multiplication make it a commutative ring. The ring of integers embeds into the ring of profinite integers by the canonical injection η : Z ↪ Z ^ , {\displaystyle \eta :\mathbb {Z} \hookrightarrow {\widehat {\mathbb {Z} }},} where n ↦ ( n mod 1 , n mod 2 , … ) . {\displaystyle n\mapsto (n{\bmod {1}},n{\bmod {2}},\dots ).} It is canonical since it satisfies the universal property of profinite groups that, given any profinite group H {\displaystyle H} and any group homomorphism f : Z → H {\displaystyle f:\mathbb {Z} \rightarrow H} , there exists a unique continuous group homomorphism g : Z ^ → H {\displaystyle g:{\widehat {\mathbb {Z} }}\rightarrow H} with f = g η {\displaystyle f=g\eta } .
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