Embedding Problems and Finite Quotients

Abstract

We give a condition on a family of solutions of quotients of an embedding problem which implies the embedding problem has a solution. This shows, in particular, that to solve an embedding problem associated to the maximal extension of a number field unramified outside a fixed finite set of places, it suffices to find a solution for each finite quotient of the embedding problem. This statement is not true in general over global function fields, but one can prove variants of it in this case in which extra conditions on the embedding problems or on the ramification of solutions are assumed