Norm tori of etale algebras and unramified Brauer groups

TL;DR

This paper provides a combinatorial description of the unramified Brauer group for varieties defined by norm equations over fields, assuming the algebra has a cyclic extension factor, advancing understanding of algebraic structures related to etale algebras.

Contribution

It introduces a new combinatorial method to describe the unramified Brauer group of norm varieties associated with etale algebras with cyclic factors.

Findings

Explicit combinatorial description of the unramified Brauer group

Applicable to norm varieties with cyclic extension factors

Enhances understanding of algebraic and arithmetic properties of norm varieties

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 $X_a$.