Profinite integer

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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

[3]

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

See also

Notes

  1. ^ Milne, Ch. I Example A. 5.
  2. ^ http://math.stackexchange.com/questions/233136/questions-on-some-maps-involving-rings-of-finite-adeles-and-their-unit-groups
  3. ^ Connes–Consani, § 2.4.
  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

External links

  • http://ncatlab.org/nlab/show/profinite+completion+of+the+integers
  • https://web.archive.org/web/20150401092904/http://www.noncommutative.org/supernatural-numbers-and-adeles/
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