Idèle
An invertible element of the ring of adèles (cf. Adèle). The set of all idèles forms a group under multiplication, called the idèle group. The elements of the idèle group of the field of rational numbers are sequences of the form
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where is a non-zero real number,
is a non-zero
-adic number,
and
for all but finitely many
(here
is the
-adic norm). A sequence of idèles
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is said to converge to an idèle if it converges to
componentwise and if there exists an
such that
for
and all
. The idèle group is a locally compact topological group in this topology. The idèle group of an arbitrary number field is constructed in an analogous way.
The multiplicative group of the field of rational numbers is isomorphically imbedded in the idèle group of this field. Every rational number is associated with the sequence
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which is an idèle. Such an idèle is said to be a principal idèle. The subgroup consisting of all principal idèles is a discrete subgroup of the idèle group.
The concepts of an idèle and an adèle were introduced by C. Chevalley in 1936 for the purposes of algebraic number theory. The new language proved useful in the study of arithmetic aspects of the theory of algebraic groups. To those ends, A. Weil generalized the definitions of an adèle and an idèle to the case of an arbitrary linear algebraic group defined over a number field.
References
[1] | A. Weil, "Basic number theory" , Springer (1973) |
[2] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986) |
Comments
Let be an index set and for each
let there be given a locally compact topological ring or group
and an open compact subring or subgroup
. The restricted direct product
of the
with respect to the
consists of all families
such that
for all but finitely many
.
becomes a locally compact group (ring) by taking as a basis of open neighbourhoods of the identity (zero) the sets
with
open in
for all
and
for all but finitely many
. For each finite set
let
. Then
is the union (direct limit) of the
.
Now let be a number field (or, more generally, a global field). Let
be the set of all prime divisors of
(both finite and infinite ones). For each
let
be the completion of
with respect to the norm of
, and let
be the ring of integers of
. (Set
if
is infinite.) Then the restricted product of the
with respect to the
is the ring of adèles
of
.
Now for each let
be the group of non-zero elements of
and let
be the group of units of
(if
is infinite take
). The restricted product of the
with respect to the
is the group of idèles of
. As a set the group of idèles
is the set of invertible elements of
. But the topology on
is stronger than that induced by
.
The quotient of by the diagonal subgroup
of principal idèles is called the idèle class group; it is important in class field theory.
The name idèle derives from ideal element. This got abbreviated id.el., which, pronounced in French, gave rise to idèle.
Idèle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Id%C3%A8le&oldid=20297