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An invertible element of the ring of adèles (cf. [[Adèle|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
+
{{MSC|11-02|11Rxx,11Sxx}}
 +
[[Category:Number theory]]
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500701.png" /></td> </tr></table>
+
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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500702.png" /> is a non-zero real number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500703.png" /> is a non-zero [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500704.png" />-adic number]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500706.png" /> for all but finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500707.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500708.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i0500709.png" />-adic norm). A sequence of idèles
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007010.png" /></td> </tr></table>
+
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
is said to converge to an idèle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007011.png" /> if it converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007012.png" /> componentwise and if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007015.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007016.png" />. 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.
+
proved useful in the study of arithmetic aspects of the theory of
 
+
algebraic groups. To those ends, A. Weil generalized the definitions
The multiplicative group of the field of rational numbers is isomorphically imbedded in the idèle group of this field. Every rational number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007017.png" /> is associated with the sequence
+
of an adele and an idele to the case of an arbitrary linear algebraic
 
+
group defined over a number field.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007018.png" /></td> </tr></table>
 
 
 
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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Weil,   "Basic number theory" , Springer (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986)</TD></TR></table>
+
<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====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007019.png" /> be an index set and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007020.png" /> let there be given a locally compact topological ring or group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007021.png" /> and an open compact subring or subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007022.png" />. The restricted direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007023.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007024.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007025.png" /> consists of all families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007027.png" /> for all but finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007028.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007029.png" /> becomes a locally compact group (ring) by taking as a basis of open neighbourhoods of the identity (zero) the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007030.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007031.png" /> open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007034.png" /> for all but finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007035.png" />. For each finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007036.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007037.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007038.png" /> is the union (direct limit) of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007039.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007040.png" /> be a number field (or, more generally, a global field). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007041.png" /> be the set of all prime divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007042.png" /> (both finite and infinite ones). For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007043.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007044.png" /> be the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007045.png" /> with respect to the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007046.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007047.png" /> be the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007048.png" />. (Set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007049.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007050.png" /> is infinite.) Then the restricted product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007051.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007052.png" /> is the ring of adèles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007054.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007055.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007056.png" /> be the group of non-zero elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007057.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007058.png" /> be the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007059.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007060.png" /> is infinite take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007061.png" />). The restricted product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007062.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007063.png" /> is the group of idèles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007064.png" />. As a set the group of idèles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007065.png" /> is the set of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007066.png" />. But the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007067.png" /> is stronger than that induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007068.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007069.png" /> by the diagonal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050070/i05007070.png" /> of principal idèles is called the idèle class group; it is important in [[Class field theory|class field theory]].
+
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 idèle derives from ideal element. This got abbreviated id.el., which, pronounced in French, gave rise to idèle.
+
The name idele derives from "ideal element". This got abbreviated
 +
"id.el.", which, pronounced in French, gave rise to "idèle".

Latest revision as of 21:21, 22 November 2014

2020 Mathematics Subject Classification: Primary: 11-02 Secondary: 11Rxx11Sxx [MSN][ZBL]

An idele (also: idèle) is an invertible element of the ring of adeles (adèles) of a global field (cf. 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, $p=2,3,5,7,\dots,$ 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

[1] A. Weil, Basic number theory, Springer (1973) | MR1344916 | Zbl 0823.11001
[2] J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967) | MR0215665 | Zbl 0153.07403


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.

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

How to Cite This Entry:
Idèle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Id%C3%A8le&oldid=11409
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article