On K\"{o}the's normality question for locally finite-dimensional central division algebras

Abstract

This paper considers K\"{o}the's question of whether every associative locally finite-dimensional (abbr., LFD) central division algebra $R$ over a field $K$ is a normally locally finite (abbr., NLF) algebra over $K$, that is, whether every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra $\mathcal{R} _{Y}$ of $R$. It shows that the answer to the posed question is negative if $K$ is a purely transcendental extension of infinite transcendence degree over an algebraically closed field $k$. On the other hand, central division LFD-algebras over $K$ turn out to be NLF in the following special cases: (i) $K$ is a finitely-generated extension of a finite or a pseudo-algebraically closed perfect field $K _{0}$; (ii) $K$ is a higher-dimensional local field with last residue field equal to $K _{0}$.