Hopf Galois structures on separable field extensions of odd prime power degree

Abstract

We study Hopf Galois structures on separable field extensions L/K of degree pn, p an odd prime, and denote by G the Galois group of the normal closure over K. We prove that if L/K has a Hopf Galois structure of cyclic type, then it has no structure of noncyclic type.For n=3, we characterize the transitive groups G such that L/K has cyclic Hopf Galois structures. We prove that if L/K has a nonabelian Hopf Galois structure of type N, then it has an abelian structure whose type has the same exponent as N. We obtain that, for p>3, the two abelian noncyclic Hopf Galois structures do not occur on the same extension. Finally, we list all possible sets of Hopf Galois structure types on a separable extension of degree p3, for p>3 a prime.