Norm tori of étale algebras and unramified Brauer groups

Abstract

AbstractLet k be a field, and let L be an étale k-algebra of finite rank. If $$a \in {k^ \times }$$ a ∈ k × , let Xa be the affine variety defined by $${N_{L/k}}(x) = a$$ 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 Xa.