Quaternion Algebras with the Same Subfields

Abstract

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.

Open problems on central simple algebras

“…Garibaldi and Saltman [GS10] showed that if F has trivial "unramified Brauer group" (in a sense including the archimedean places), then any two quaternion division algebras over F having the same maximal subfields must be isomorphic. Rapinchuk and Rapinchuk proved similar results for period 2 division algebras in [RR09].…”

Division algebras with the same maximal subfields

“…Lemma 2.3 of [40] (see also [20], Lemma 3.1) now shows that for any set V of discrete valuations of K, we have gen(D) ⊂ gen V (D).…”

“…Several people, including Garibaldi, Rost, Saltman, Schacher, Wadsworth, and others have given a construction of quaternion algebras with non-trivial genus over certain very large fields of characteristic ̸ = 2. 2 We refer the reader to [20], § 2 for the full details, and only sketch the main ideas here.…”

“…In connection with their work on length-commensurable locally symmetric spaces in [36], Prasad and the second-named author of this paper asked whether |gen(D)| = 1 for any central quaternion division algebra D over K = Q(x). 3 This question was answered in the affirmative by Saltman, and later, in a joint paper with Garibaldi, it was shown that any quaternion division algebra over k(x), where k is an arbitrary number field, has trivial genus (in fact, they considered more generally so-called transparent fields of characteristic ̸ = 2; see [20]). Motivated by this result, we showed in [40] that for a given field k of characteristic ̸ = 2 the triviality of the genus for quaternion division algebras over k is a property that is stable under purely transcendental extensions.…”

Division algebras with infinite genus

“…However, for even finitely generated fields, there are open questions. In [3], Garibaldi and Saltman produce an infinitely generated field over which there are two nonisomorphic weakly isomorphic quaternion division algebras. With this in mind define the genus of D to be the set:…”