A useful proposition for division algebras of small degree

Abstract

Let F F be a field and let D D be an F F -central division algebra of degree n n . We present a short, elementary proof of the following statement: There is an n − 1 n - 1 -dimensional F F -subspace V V of D D such that for every nonzero element ν \nu of V V , Tr ( ν ) = Tr( ν − 1 ) = 0 {\text {Tr}}(\nu ) = {\text {Tr(}}{\nu ^{ - 1}}) = 0 . We then indicate how one can use this result to obtain the basic structural results on division algebras of degree three and four (results of Wedderburn and Albert, respectively).