Concept
In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) where indicates the profinite completion of , the index runs over all prime numbers, and 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. The profinite integers can be constructed as the set of sequences of residues represented as such that . Pointwise addition and multiplication make it a commutative ring. The ring of integers embeds into the ring of profinite integers by the canonical injection: where It is canonical since it satisfies the universal property of profinite groups that, given any profinite group and any group homomorphism , there exists a unique continuous group homomorphism with . Every integer has a unique representation in the factorial number system as where for every , and only finitely many of are nonzero. Its factorial number representation can be written as . In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string , where each is an integer satisfying . The digits determine the value of the profinite integer mod . More specifically, there is a ring homomorphism sending The difference of a profinite integer from an integer is that the "finitely many nonzero digits" condition is dropped, allowing for its factorial number representation to have infinitely many nonzero digits. Another way to understand the construction of the profinite integers is by using the Chinese remainder theorem. Recall that for an integer with prime factorization of non-repeating primes, there is a ring isomorphism from the theorem. Moreover, any surjection will just be a map on the underlying decompositions where there are induced surjections since we must have . It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism with the direct product of p-adic integers.