Sums of corestrictions of cyclic algebras

By a cyclic layer of a finite Galois extension,E/K, of fields one means a cyclic extension,L/F, of fields whereE⊇L⊇F⊇K. LetC(E/K) denote the subgroup of the relative Brauer group, Br(E/K), generated by the various subgroups cor(Br(L/F)) asL/F ranges over all cyclic layers ofE/K and where cor denotes the corestriction map into Br(E/K). We show that forK global, Br(E/K) :C(E/K)]<∞ and we produce examples whereC(E/K)≠Br(E/K). In memory of S.A. Amitsur, our teacher, friend, collaborator, and inspiration. Supported in part by NSA Grant No. MDA904-95-H-1054. Supported in part by NSA Grant No. MDA904-95-H-1022.