Idele-to-ideal homomorphism
= Idele-to-ideal homomorphism
{title2=$(x_v)_v\mapsto\prod_{v<\infty}\mathfrak p_v^{\operatorname{ord}_v(x_v)}$}
The map from an <idele> to its finite-place <fractional ideal> is surjective, has kernel the product of the infinite multiplicative groups and finite-place unit groups, and sends diagonal elements to <principal fractional ideals>. It realizes the <ideal class group> as the corresponding quotient of the <idele group>.