R. Hazrat, A. Wadsworth: Nontriviality of NK1 for division algebras

A field $E$ is said to be NKNT if for any noncommutative division algebra $D$ finite dimensional over $E\subseteq Z(D)=F$ with index $\ind(D)$, $\Nrd(D^*)/F^{*\ind(D)}$ is nontrivial. It is proved that if $E$ is a field finitely generated but not algebraic over some subfield then $E$ is NKNT. As a consequence, if $F=Z(D)$ is finitely generated over its prime subfield or over an algebraically closed field, then $\CK(D)=\text{Coker}(K_1 F \rightarrow K_1 D)$ is nontrivial.