# Idèle

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.