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

Abstract

Abstract Let D be a division algebra over its center F of degree n. Consider the group μ Z ⁢ ( D ) = μ n ⁢ ( F ) / Z ⁢ ( D ′ ) {\mu_{Z}(D)=\mu_{n}(F)/Z(D^{\prime})} , where μ n ⁢ ( F ) {\mu_{n}(F)} is the group of all the n-th roots of unity in F * {F^{*}} , and Z ⁢ ( D ′ ) {Z(D^{\prime})} is the center of the commutator subgroup of the group of units D * {D^{*}} of D. It is shown that if μ Z ⁢ ( D ⊗ F L ) ≠ 1 {\mu_{Z}(D\otimes_{F}L)\neq 1} for some L containing all the primitive n k {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.