Hopf-Galois structures on separable field extensions of degree pq

Abstract

In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions L/K is a natural next step. One must consider now the interplay between two Galois groups G=Gal(E/K) and G′=Gal(E/L), where E is the Galois closure of L/K. In this paper, we give a characterisation and enumeration of the Hopf-Galois structures arising on separable extensions of degree pq where p and q are distinct odd primes. This work includes the results of Byott and Martin-Lyons who do likewise for the special case that p=2q+1.