Hopf–Galois structures on Galois field extensions of degree pq

Abstract

We determine all Hopf–Galois structures on a Galois extension of fields of degree pq, where p, q are primes with $\text{p≡1}\text{(}\text{mod}\text{q)}$. There are 2q−1, respectively 2+p(2q−3), Hopf–Galois structures when the extension is cyclic, respectively nonabelian. Explicit generators are given for the groups of permutations corresponding to these Hopf–Galois structures.