Hopf-Galois structures on separable field extensions of degree pq

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 = 2 q + 1. (c) 2024 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).