Brauer Factor Sets and Simple Algebras

Abstract

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 ${c_{ijk}} = c_{kji}^{ - 1}$. This implies at once the existence of an element $a \ne 0$ with ${\text {tr}}(a) = {\text {tr}}({a^2}) = 0$; the coefficients of ${x^{n - 1}}$ and ${x^{n - 2}}$ in 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.