Quadratic Descent for Quaternion Algebras

Abstract

Given a Galois embedding problem H 3 G GalL=F with kernel of order two and F Hilbertian, we consider how obstructions O K to subgroup embedding problems H 0 3 G 0 GalL=K for K : F 2 descend to the obstruction O F to the original embedding problem, up to Br 2 K=F. In particular, to such an obstruction O K we associate a tower of Z=2Z-embedding problems and prove that the contribution of O K to O F is given by the obstruction to the last embedding problem in the tower. We show that such an association in fact holds generally for central, Brauer Z=pZ-problems. When O K is the class of a quaternion algebra, we give an explicit representation of O F up to Br 2 K=F. As a consequence, we represent in terms of quaternion algebras over F the obstructions to Z=2Z Â Z=2Z-embedding problems not previously determined over F. 4523