On the residue fields of Henselian valued stable fields, II

TL;DR

This paper investigates the structure of residue fields in Henselian valued stable fields, focusing on Galois extensions, division algebras, and character groups, revealing new embedding limitations and properties of multiplicative groups.

Contribution

It extends previous results by showing non-embedding conditions for Galois extensions in division algebras and characterizes the multiplicative group structure of quasilocal fields.

Findings

Non-nilpotent Galois groups do not guarantee embedding into division algebras.

Character group of absolute Galois group of quasilocal fields is explicitly described.

Divisible part of the multiplicative group equals intersection of norm groups in almost perfect quasilocal fields.

Abstract

Let $E$ be a primarily quasilocal field, $M/E$ a finite Galois extension and $D$ a central division $E$-algebra of index divisible by $[M\colon E]$. In addition to the main result of Part I, this part of the paper shows that if the Galois group $G(M/E)$ is not nilpotent, then $M$ does not necessarily embed in $D$ as an $E$-subalgebra. When $E$ is quasilocal, we find the structure of the character group of its absolute Galois group; this enables us to prove that if $E$ is strictly quasilocal and almost perfect, then the divisible part of the multiplicative group $E ^{\ast}$ equals the intersection of the norm groups of finite Galois extensions of $E$.