Norm tori of etale algebras and unramified Brauer groups

Abstract

Let $k$ be a field, and let $L$ be an \'etale k-algebra of finite rank. If $a$ is a nonzero element in $k$, let $X_a$ be the affine variety defined by the norm equation $N_{L/k}(x) = a$. Assuming that $L$ has at least one factor that is a cyclic field extension of $k$, we give a combinatorial description of the unramified Brauer group of $X_a$.