Profinite integer
Appearance
In mathematics, a profinite integer is an element of the ring
where p runs over all prime numbers, is the ring of p-adic integers and (profinite completion).
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 Pontrjagin dual of .
See also
Notes
- ^ Milne, Ch. I Example A. 5.
- ^ http://math.stackexchange.com/questions/233136/questions-on-some-maps-involving-rings-of-finite-adeles-and-their-unit-groups
- ^ Connes–Consani, § 2.4.
- ^ K. Conrad, The character group of Q
References
- Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580.
- Milne, Class Field Theory