The tame-wild principle for discriminant relations for number fields

Consider tuples (K₁,..., K_r)of separable algebras over a common local or global number field F, with the K_i related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants of K_i =F. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.