View source for Idèle

{{MSC|11-02|11Rxx,11Sxx}}
[[Category:Number theory]]

An idele (also: idèle) is an invertible element of the ring of 
adeles (adèles) of a global field (cf.
[[Adèle|Adele]]). The set of all ideles forms a group under
multiplication, called the idele group. The elements of the idele
group of the field of rational numbers are sequences of the form 
$$a = (a_\infty,a_2,\dots,a_p,\dots),$$
where $a_\infty$ is a non-zero real number, $a_p$ is a non-zero
[[P-adic number|$p$-adic number]], $p=2,3,5,7,\dots,$ and $|a_p|=1$ for all but finitely
many $p$ (here $|x|_p$ is the $p$-adic norm). A sequence of ideles 
$$a^{(n)} = (a_\infty^{(n)},a_2^{(n)},\dots,a_p^{(n)},\dots),$$
is said to converge to an idele $a$ if it converges to $a$
componentwise and if there exists an $N$ such that $|a_p^{-1}a_p^{(n)}|_p = 1$ for $n>N$ and all
$p$. The idele group is a locally compact topological group in this
topology. The idele 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 idele group of this field. Every
rational number $r\ne 0$ is associated with the sequence 
$$(r,r,\dots,r,\dots),$$
which is an
idele. Such an idele is said to be a principal idele. The subgroup
consisting of all principal ideles is a discrete subgroup of the idele
group.

The concepts of an idele and an adele 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 adele and an idele to the case of an arbitrary linear algebraic
group defined over a number field.

====References====
<table><TR><TD valign="top">[1]</TD>
 <TD valign="top"> A. Weil, ''Basic number theory'', Springer (1973)
 | {{MR|1344916}} | {{ZBL|0823.11001}} </TD>
</TR><TR><TD valign="top">[2]</TD>
 <TD valign="top"> J.W.S. Cassels (ed.)  A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press (1967) | {{MR|0215665}}  |  {{ZBL|0153.07403}}</TD>
</TR></table>



====Comments====
Let $I$ be an index set and for each $i\in I$ let there be
given a locally compact topological ring or group $G_i$ and an open
compact subring or subgroup $B_i$. The restricted direct product $G=\Pi' G_i$ of
the $G_i$ with respect to the $B_i$ consists of all families $(g_i)_{i\in I}$ such that
$g_i\in B_i$ for all but finitely many $i$. $G$ becomes a locally compact group
(ring) by taking as a basis of open neighbourhoods of the identity
(zero) the sets $\prod_i U_i$ with $U_i$ open in $G_i$ for all $i$ and $U_i = B_i$ for all
but finitely many $i$. For each finite set $S\subset I$ let $G_S = \prod_{i\in S} \times \prod_{i\notin S} B_i$. Then $G$ is
the union (direct limit) of the $G_S$.

Now let $k$ be a number field (or, more generally, a global
field). Let $I$ be the set of all prime divisors of $k$ (both finite
and infinite ones). For each $\def\fp{\mathfrak{p}} \fp\in I$ let $k_\fp$ be the completion of $k$ with
respect to the norm of $\fp$, and let $A_\fp$ be the ring of integers of
$k_\fp$. (Set $A_\fp = k_\fp$ if $\fp$ is infinite.) Then the restricted product of the
$k_\fp$ with respect to the $A_\fp$ is the ring of adeles $A_k$ of $k$.

Now for each $\fp\in I$ let $k_\fp^*$ be the group of non-zero elements of $k_\fp$ and
let $U_\fp$ be the group of units of $k_\fp^*$ (if $\fp$ is infinite take
$U_\fp = k_\fp^*$). The restricted product of the $k_\fp^*$ with respect to the $U_\fp$ is the
group of ideles of $k$. As a set the group of ideles $I_k$ is the set of
invertible elements of $A_k$. But the topology on $I_k$ is stronger than
that induced by $A_k$.

The quotient of $I_k$ by the diagonal subgroup $k^* = \{(\alpha)_{i\in I}\}$ of principal ideles
is called the idele class group; it is important in
[[Class field theory|class field theory]].

The name idele derives from "ideal element". This got abbreviated
"id.el.", which, pronounced in French, gave rise to "idèle".