Division algebras over \(C_{2}\)- and \(C_{3}\)-fields (Q2781327)

A field \(F\) is called a \(C_m\)-field for some \(m\) if every form of degree \(d\) in \(n>d^m\) variables has a nontrivial zero. The author proves by elementary methods that any division algebra of degree \(4\) over a \(C_3\)-field containing \(\sqrt {-1}\) and that any division algebra of degree \(8\) over a \(C_2\)-field containing \(\root 4\of{-1}\) is cyclic. The degree \(4\) case already follows from a theorem of Rost, Serre and Tignol.