The index of a Brauer class on a Brauer-Severi variety

Abstract: Let D and E be central division algebras over k; let K be the generic splitting field of E; we show that the index of D ⊗k K is the minimum of the indices of D ⊗ E⊗i as i varies. We use this to calculate the index of D under related central extensions and to construct division algebras with special properties. A question that seems not to have been discussed is the following. Let D and E be central division algebras over a field k; let K be the function field of the Brauer-Severi variety, Br-S(E), what is the index of D⊗kK? It is this question that we shall answer. We show that it is the minimum of the indices of D⊗E⊗