Nontriviality of certain quotients of K₁ groups of division algebras

For a division algebra D finite dimensional over its center Z (D) = F, it is a conjecture that CK₁ (D) : = Coker (K₁ F → K₁ D) is trivial if and only if D ≅ (frac(- 1, - 1, F)) with F a formally real Pythagorean field. Since CK₁ (D) is very difficult to work with, we consider here NK₁ (D) : = Nrd_D (D^*) / F^{* ind (D)}, which is a homomorphic image of CK₁ (D). A field E is said to be NKNT if for every noncommutative division algebra D finite dimensional over E ⊆ Z (D), NK₁ (D) is nontrivial. It is proved that if E is finitely generated but not algebraic over some subfield then E is NKNT. As a consequence, if Z (D) is finitely generated over its prime subfield or over an algebraically closed field, then CK₁ (D) is nontrivial.