Quaternion algebras with the same subfields

TL;DR

This paper investigates conditions under which quaternion division algebras with identical subfields are isomorphic, establishing positive results for certain field extensions and exploring related algebraic structures.

Contribution

It proves that quaternion division algebras over specific field extensions are isomorphic if they share the same subfields, extending known results and applying to Pfister forms.

Findings

Quaternion algebras with the same subfields are isomorphic over certain global field extensions.

The result applies to fields where the extension is unirational with zero unramified Brauer group.

A similar isomorphism result is established for Pfister forms.

Abstract

G. Prasad and A. 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.