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

Abstract

Let D D and E E be central division algebras over k k ; let K K be the generic splitting field of E E ; we show that the index of D ⊗ k K D{ \otimes _k}K is the minimum of the indices of D ⊗ E ⊗ i D \otimes {E^{ \otimes i}} as i i varies. We use this to calculate the index of D D under related central extensions and to construct division algebras with special properties.