On division algebras having the same maximal subfields

Abstract

We show that if a field K of characteristic ≠ 2 satisfies the following property (*) for any two central quaternion division algebras D 1 and D 2 over K, the fact that D 1 and D 2 have the same maximal subfields implies that D 1 ≃ D 2 over K, then the field of rational functions K(x) also satisfies (*). This, in particular, provides an alternative proof for the result of S. Garibaldi and D. Saltman that the fields of rational functions k(x 1, . . . , x r ), where k is a number field, satisfy (*). We also show that K = k(x 1, . . . , x r ), where k is either a totally complex number field with a single dyadic place (e.g. $${k = \mathbb{Q}(\sqrt{-1})}$$ ) or a finite field of characteristic ≠ 2, satisfies the analog of (*) for all central division algebras having exponent two in the Brauer group Br(K).