Brauer factor sets and simple algebras

It is shown that the Brauer factor set (c_ijk) of a finite-dimensional division algebra of odd degree n can be chosen such that c_iji = c_iij =c_jii =1 for all i, j and C_ijkThis implies at once the existence of an element a 0 with tr (a)=tr(a²)=0; the coefficients of x^n-1 and x^n-2in the characteristic polynomial of a are thus 0. Also one gets a generic division algebra of degree n whose center has transcendence degree n+(n-1)(n-2)/2 as well as a new (simpler) algebra of generic matrices. Equations are given to determine the cyclicity of these algebras, but they may not be tractable.