Hopf–Galois structures on Galois field extensions of degree pq

We determine all Hopf–Galois structures on a Galois extension of fields of degree pq, where p, q are primes with p≡1 (mod 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.