On Fields of dimension one that are Galois extensions of a global or local field

TL;DR

This paper characterizes when the Brauer group of Galois extensions over global or local fields is trivial, linking it to properties of residue fields, valuation extensions, and tame abelian extensions, and discusses violations of the Hasse principle.

Contribution

It provides new criteria for trivial Brauer groups in Galois extensions over local and global fields, including explicit conditions and examples of Hasse principle violations.

Findings

Br(E) = 0 characterized by residue field properties over local fields.

Fields with Br(E) = 0 in global case are tame abelian extensions.

Existence of forms violating the Hasse principle for any degree n.

Abstract

Let $K$ be a global or local field, $E/K$ a Galois extension, and Br$(E)$ the Brauer group of $E$. This paper shows that if $K$ is a local field, $v$ is its natural discrete valuation, $v'$ is the valuation of $E$ extending $v$, and $q$ is the characteristic of the residue field $\widehat E$ of $(E, v')$, then Br$(E) = \{0\}$ if and only if the following conditions hold: $\widehat E$ contains as a subfield the maximal $p$-extension of $\widehat K$, for each prime $p \neq q$; $\widehat E$ is an algebraically closed field in case the value group $v'(E)$ is $q$-indivisible. When $K$ is a global field, it characterizes the fields $E$ with Br$(E) = \{0\}$, which lie in the class of tame abelian extensions of $K$. We also give a criterion that, in the latter case, for any integer $n \ge 2$, there exists an $n$-variate $E$-form of degree $n$, which violates the Hasse principle.