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

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.