Discreteness of principal ideles
= Discreteness of principal ideles
The diagonal multiplicative group of a <number field> is discrete in its <idele group>. Require local units at every finite place and an archimedean neighbourhood of one. A diagonal element $x$ meeting these restrictions has integral $x-1$. If $x\ne1$, its nonzero integer <field norm> cannot have absolute value less than one, while sufficiently small archimedean differences would force exactly that.