Relative Embedding Problems

Abstract:

We consider Galois embedding problems $G\twoheadrightarrow H\cong \operatorname {Gal}(X/Z)$ such that a Galois embedding problem $G\twoheadrightarrow \operatorname {Gal}(Y/Z)$ is solvable, where $Y/Z$ is a Galois subextension of $X/Z$. For such embedding problems with abelian kernel, we prove a reduction theorem, first in the general case of commutative $k$-algebras, then in the more specialized field case. We demonstrate with examples of dihedral embedding problems that the reduced embedding problem is frequently of smaller order. We then apply these results to the theory of obstructions to central embedding problems, considering a notion of quotients of central embedding problems, and classify the infinite towers of metacyclic $p$-groups to which the reduction theorem applies.