Cyclotomic splitting fields for projective Schur algebras over number fields

Let k be a number field and for every positive integer m denote by ξm a root of unity of order m in an algebraically closed field extension C of k. Let A be a simple k-algebra wich occurs as a simple component of a twisted group algebra (k,G, f) where G is a finite group and f : G × G → k* is a central 2-cocycle. Denote by K ⊂ C the center of A. The purpose of this note is to construct a cyclotomic splitting field for A of the form K(ξm) where m is determined from properties of K,f and of a certain simple component of the ordinary group algebra of some finite central group extension of the commutator subgroup of G. The proof is based on results in representation theory, in cohomology and in the theory of algebras which are all well known.