Hopf-Galois structures on separable field extensions of degree related to Cunningham chains

TL;DR

This paper investigates Hopf-Galois structures on separable field extensions of squarefree degree, especially those where consecutive primes are related by a specific doubling-plus-one relation, extending previous work on such algebraic structures.

Contribution

It generalizes prior results to include a broader class of squarefree degrees with prime factors related by a specific recursive relation.

Findings

Extended the classification of Hopf-Galois structures to new prime-related degrees.

Identified conditions under which certain Hopf-Galois structures exist for these extensions.

Provided new insights into the algebraic structure of extensions with prime factors linked by p=2q+1.

Abstract

The past few years have seen Hopf--Galois structures on extensions of squarefree degree studied in various contexts. The Galois case was fully explored by Alabdali and Byott in 2020, followed by a first attempt at generalising these results to include non-normal extensions by Byott and Martin-Lyons; their work looks at separable extensions of degree $pq$ with $p,q$ distinct odd primes, and $p=2q+1$. This paper extends the latter work further by considering separable extensions of squarefree degree $n=p_1...p_m$ where each pair of consecutive primes $p_i,p_{i+1}$ are related by $p_i=2p_{i+1}+1$.