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 $\hat{\Delta }$ if and only if the degree $[\hat{R}:\hat{K}]$ divides the index of $\hat{\Delta }$.

Final form, to appear in J. Algebra 319 (2008), No 1, 16-49

References in corpus (2)

• Algebraic extensions of global fields admitting one-dimensional local class field theory
• On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions

Cited by in corpus (8)

• On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions
• Primarily quasilocal fields and 1-dimensional abstract local class field theory
• On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions
• Demushkin groups and inverse Galois theory for pro-p-groups of finite rank and maximal p-extensions
• Fields of dimension one algebraic over a global or local field need not be of type $C_1$
• On Fields of dimension one that are Galois extensions of a global or local field
• On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions
• Norm groups and class fields of formally real quasilocal fields