Relative Brauer groups of discrete valued fields

Abstract:

Let $E$ be a non-trivial finite Galois extension of a field $K$. In this paper we investigate the role that valuation-theoretic properties of $E/K$ play in determining the non-triviality of the relative Brauer group, $\operatorname {Br} (E/K)$, of $E$ over $K$. In particular, we show that when $K$ is finitely generated of transcendence degree 1 over a $p$-adic field $k$ and $q$ is a prime dividing $[E:K]$, then the following conditions are equivalent: (i) the $q$-primary component, $\operatorname {Br} (E/K)_{q}$, is non-trivial, (ii) $\operatorname {Br} (E/K)_{q}$ is infinite, and (iii) there exists a valuation $\pi$ of $E$ trivial on $k$ such that $q$ divides the order of the decomposition group of $E/K$ at $\pi$.