On the non-existence of cyclic splitting fields for division algebras

Let D be a division algebra over its center F of degree n. Consider the group mu(Z)(D) = mu(n)(F)/Z(D'), where mu(n)(F) is the group of all the n-th roots of unity in F*, and Z(D') is the center of the commutator subgroup of the group of units D* of D. It is shown that if mu(Z)(D circle times(F) L) not equal 1 for some L containing all the primitive n(k)-th roots of unity for all positive integers k, then D is not split by any cyclic extension of F. This criterion is employed to prove that some special classes of division algebras are not cyclically split.