Classification of the Hopf Galois Structures on Prime Power Radical Extensions

Abstract

Letpbe an odd prime andna positive integer and letkbe a field of characteristic zero. LetK=k(w) withwpn=a∈kwhereais such that [K:k]=pn and letrdenote the largest integer between 0 andnsuch thatK∩k(ζpr)=k(ζpr), where ζpr denotes a primitivepr th root of unity. The extensionK/kis separable, but not necessarily normal and, by Greither and Pareigis, isH-Galois withHaK̃-Hopf algebra form of a group ringkNwhereK̃is the normal closure ofK/k.His said to bealmost classicalifN