In mathematics, a profinite integer is an element of the ring
Z ^ = ∏ p Z p where p runs over all prime numbers, Z p is the ring of p-adic integers and Z ^ = lim ← Z / n Z (profinite completion).
Example: Let F ¯ q be the algebraic closure of a finite field F q of order q. Then Gal ( F ¯ q / F q ) = Z ^ .
A usual (rational) integer is a profinite integer since there is the canonical injection
Z ↪ Z ^ , n ↦ ( n , n , … ) . The tensor product Z ^ ⊗ Z Q is the ring of finite adeles A Q , f = ∏ p ′ Q p of Q where the prime ' means restricted product.
There is a canonical paring
Q / Z × Z ^ → U ( 1 ) , ( q , a ) ↦ χ ( q a ) where χ is the character of A Q , f induced by Q / Z → U ( 1 ) , α ↦ e 2 π i α . The pairing identifies Z ^ with the Pontrjagin dual of Q / Z .
