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, $v^{\prime }$ is the valuation of $E$ extending $v$, and $q$ is the characteristic of the residue field $\hat{E}$ of $(E,v^{\prime })$, then Br$(E)=\{0\}$ if and only if the following conditions hold: $\hat{E}$ contains as a subfield the maximal $p$-extension of $\hat{K}$, for each prime $p\ne q$; $\hat{E}$ is an algebraically closed field in case the value group $v^{\prime }(E)$ 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.

LaTeX, 18 pages, no figures

References in corpus (2)

• Autour d'une conjecture de Kato et Kuzumaki
• Fields of cohomological dimension one versus C_1-fields