Profinite integer
In mathematics, a profinite integer is an element of the ring
where indicates the profinite completion of, the index p runs over all prime numbers, and is the ring of p-adic integers.
Concretely the profinite integers will be the set of maps such that and. Pointwise addition and multiplication makes it a commutative ring. If a sequence of integers converges modulo n for every n then the limit will exist as a profinite integer.
Example: Let be the algebraic closure of a finite field of order q. Then.
A usual integer is a profinite integer since there is the canonical injection
The tensor product is the ring of finite adeles of where the prime ' means restricted product.
There is a canonical pairing
where is the character of induced by. The pairing identifies with the Pontryagin dual of.
