Hopf-Galois Structures on Separable Field Extensions of Squarefree Degree

Abstract

In 1987, Greither and Pareigis showed that the problem of finding Hopf- Galois structures on separable extensions can be translated to a problem in group theory. This has since opened the door to a plethora of literature and research into separable Hopf-Galois theory, which has in turn allowed for several classification results as well as connections with other areas of mathematics to be subsequently discovered. In this thesis, we present a series of classification studies and results which mainly look at Hopf-Galois structures on certain classes of separable field extensions of squarefree degree. The majority of the work follows methods developed by Byott, which show that the groups of interest in the approach of Greither and Pareigis relate to transitive subgroups of the holomorph of another group.