Hopf-Galois structures on non-normal extensions of degree related to Sophie Germain primes

Abstract

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions L/K of squarefree degree n. If E/K is the normal closure of L/K then G = Gal(E/K) can be viewed as a permutation group of degree n. We show that G has derived length at most 4, but that many permutation groups of squarefree degree and of derived length 2 cannot occur. We then investigate in detail the case where n = pq where q ≥ 3 and p = 2q + 1 are both prime. (Thus q is a Sophie Germain prime and p is a safeprime). We list the permutation groups G which can arise, and we enumerate the Hopf-Galois structures for each G. There are six such G for which the corresponding field extensions L/K admit Hopf-Galois structures of both possible types.