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

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 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 field; (ii) K is a higher-dimensional local field with a finite last residue field.