Hopf-Galois structures on parallel extensions

TL;DR

This paper explores the relationship between Hopf-Galois structures on a finite separable extension and its parallel subextensions, revealing conditions under which such structures exist or do not exist, especially for degree pq extensions.

Contribution

It introduces the concept of parallel extensions in Hopf-Galois theory and analyzes their impact on the existence of Hopf-Galois structures, including infinite families and specific degree cases.

Findings

Existence of transitive subgroups with differing Hopf-Galois structures on parallel extensions.

Extension of such phenomena into infinite families of subgroups.

No examples of degree pq extensions with this phenomenon for distinct odd primes.

Abstract

Let $L/K$ be a finite separable extension of fields of degree $n$, and let $E/K$ be its Galois closure. Greither and Pareigis showed how to find all Hopf--Galois structures on $L/K$. We will call a subextension $L'/K$ of $E/K$ \textit{parallel} to $L/K$ if $[L':K]=n$. In this paper, we investigate the relationship between the Hopf--Galois structures on an extension $L/K$ and those on the related parallel extensions. We give an example of a transitive subgroup corresponding to an extension admitting a Hopf--Galois structure but that has a parallel extension admitting no Hopf--Galois structures. We show that once one has such a situation, it can be extended into an infinite family of transitive subgroups admitting this phenomenon. We also investigate this fully in the case of extensions of degree $pq$ with $p,q$ distinct odd primes, and show that there is no example of such an extension admitting the phenomenon.