Cohomological operations for the Brauer group and Witt kernels of a quartic field extension

Abstract

Let [Formula: see text] be a field, [Formula: see text], [Formula: see text][Formula: see text], [Formula: see text] a quartic field extension. We investigate the divided power operation [Formula: see text] on the group [Formula: see text]. In particular, we show that any element of [Formula: see text] is a symbol [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] is the quadratic trace form associated with the extension [Formula: see text]. As an application, we obtain certain results on the Stifel–Whitney maps [Formula: see text].