Normalized Brauer Factor Sets

Abstract

We investigate normalized Brauer factor sets of central simple algebras with respect to arbitrary maximal separable subalgebras, and show that they have a cohomological description. As a consequence, a central simple algebra of even degree having a normalized Brauer factor set cannot be a division algebra. An intrinsic equivalent condition is given for a central simple algebra to have a normalized Brauer factor set. Consequently, an algebra has a normalized Brauer factor set if it is a square in the relative Brauer group. The converse holds for index 4, or for symbols, but an example is given of an algebra of index 8 with normalized Brauer factor set, which isnota square in the relative Brauer group. On the other hand, supposeDis a division algebra of odd degree. IfDhas a maximal separable subfieldKwhose Galois groupGsatisfies a certain property (which automatically holds for |G| odd) thenDcontains an elementafor which tr(a)=tra2=tra−1=0.