Relative Embedding Problems

Abstract

Abstract. We consider Galois embedding problems G H ∼ = Gal(X/Z) such that a Galois embedding problem G 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.

Quadratic Descent for Quaternion Algebras

“…However, the representation of obstructions to other Z=2Z-embedding problems as tensor products of quaternion algebras has required a variety of other computational techniques. The most recent survey of results and techniques in the theory of Z=2Z-embedding problems appeared in 1995 [8], and later results are due to Black-Swallow [4] and Ledet [15,16].…”