Weak approximation for number fields

Given finitely many distinct places of a number field $v_1,\dots ,v_r$, the diagonal image of $K$ is dense in ${\prod }_i{K}_{v_i}$. One can therefore simultaneously prescribe finite congruences and real signs. This is useful in the ray class exact sequence and in finding representatives of an ideal class prime to a prescribed finite set of prime ideals.

At finitely many finite places of a number field, choose square-class parameters for the local unramified quadratic extensions. Local square classes are open, including at dyadic places by the strong form of Hensel lemma. Weak approximation for number fields gives a global parameter in all the prescribed local classes. Its quadratic extension is unramified of residue degree two at those places, so their extended prime ideals remain prime. If no finite place is prescribed, an auxiliary local nonsquare condition ensures the global extension is nontrivial.