Hopf-Galois Structures on Non-Normal Extensions of Degree Related to Sophie Germain Primes

TL;DR

This paper classifies Hopf-Galois structures on certain non-normal field extensions of squarefree degree, especially when the degree is a product of a Sophie Germain prime and a safe prime, revealing the possible Galois groups and structures.

Contribution

It provides a detailed classification of Hopf-Galois structures for extensions of degree pq with p=2q+1 prime, including explicit enumeration of possible Galois groups and structures.

Findings

Galois group G has derived length at most 4.

Many groups of squarefree degree and derived length 2 do not occur.

Six groups admit both types of Hopf-Galois structures.

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 $G=\mathrm{Gal}(E/K)$ 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 $q \geq 3$ 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.