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
.
