Quaternion algebras with the same subfields

Abstract

Prasad and Rapinchuk asked if two quaternion division F-algebras that have the same subfields are necessarily isomorphic. The answer is known to be "no" for some very large fields. We prove that the answer is "yes" if F is an extension of a global field K so that F/K is unirational and has zero unramified Brauer group. We also prove a similar result for Pfister forms and give an application to tractable fields. © Springer Science+Business Media, LLC 2010.