Nontriviality of certain quotients of K1 groups of division algebras

Abstract

For a division algebra D finite dimensional over its center Z(D) = F , it is a conjecture that CK 1 (D) := Coker(K 1 F → K 1 D) is trivial if and only if D ∼ = ( −1,−1 F ) with F a formally real Pythagorean field. Since CK 1 (D) is very difficult to work with, we consider here NK 1 (D) := Nrd D (D * )/F * ind(D) , which is a homomorphic image of CK 1 (D). A field E is said to be NKNT if for every noncommutative division algebra D finite dimensional over E ⊆ Z(D), NK 1 (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 CK 1 (D) is nontrivial.