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 (non-integral) 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 .[1]

A usual (rational) 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.[2]

There is a canonical pairing


where is the character of induced by .[4] The pairing identifies with the Pontryagin dual of .

See also



  • Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580.
  • Milne, Class Field Theory
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