Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power

Abstract

A Hopf Galois structure on a finite field extension [Formula: see text] is a pair [Formula: see text], where [Formula: see text] is a finite cocommutative [Formula: see text]-Hopf algebra and [Formula: see text] a Hopf action. In this paper, we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree [Formula: see text], with [Formula: see text] an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree [Formula: see text], we determine exactly the possible sets of Hopf Galois structure types.