Nontriviality of certain quotients of K1 groups of division algebras

AbstractFor a division algebra D finite dimensional over its center Z(D)=F, it is a conjecture that CK1(D):=Coker(K1F→K1D) is trivial if and only if D≅(−1,−1F) with F a formally real Pythagorean field. Since CK1(D) is very difficult to work with, we consider here NK1(D):=NrdD(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