Consider tuples (K₁,…,K_r) of separable algebras over a common local or global number field F 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.