The second Stiefel–Whitney class of the quadratic form
trA/k(ax2) is computed, where A is a
central simple algebra over a perfect field k of characteristic different from 2,
a ∈ A is a fixed element, and trA/k is the reduced trace.
This class is related on the one hand to the class of A in the Brauer group, and on the other hand to
corestrictions of quaternion algebras over certain factors arising from E[otimes ]kE,
where E is a commutative étale algebra over k that depends on the semisimple part of a.