On the residue fields of Henselian valued stable fields

Abstract

Let $(K, v)$ be a Henselian valued field satisfying the following conditions, for a given prime number $p$: (i) central division $K$-algebras of (finite) $p$-primary dimensions have Schur indices equal to their exponents; (ii) the value group $v(K)$ properly includes its subgroup $pv(K)$. The paper shows that if $\hat K$ is the residue field of $(K, v)$ and $\hat R$ is an intermediate field of the maximal $p$-extension $\hat K (p)/\hat K$, then the natural homomorphism Br$(\hat K) \to $ Br$(\hat R)$ of Brauer groups maps surjectively the $p$-component Br$(\hat K)_{p}$ on Br$(\hat R)_{p}$. It proves that Br$(\hat K)_{p}$ is divisible, if $p > 2$ or $\hat K$ is a nonreal field, and that Br$(\hat K)_{2}$ is of order 2 when $\hat K$ is formally real. We also obtain that $\hat R$ embeds as a $\hat K$-subalgebra in a central division $\hat K$-algebra $\hat Δ$ if and only if the degree $[\hat R\colon \hat K]$ divides the index of $\hat Δ$.