 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 $x\ne 1$, its nonzero integer field norm cannot have absolute value less than one, while sufficiently small archimedean differences would force exactly that.