Hopf-Galois structures on parallel extensions

TL;DR

This work analyzes how Hopf–Galois structures on a separable extension $L/K$ interact with parallel degree-$n$ subextensions inside its Galois closure. Using Byott’s holomorph perspective, it reduces the problem to classifying transitive subgroups of $\mathrm{Hol}(N)$ and studying index-$n$ subgroups via their cores, enabling precise results for degree $pq$ extensions with $p>q$ odd primes. It shows that for $n=pq$, parallel extensions inherit Hopf–Galois structures of the same type as the original extension: cyclic type in the cyclic case and non-abelian type in the non-abelian case, with a detailed classification of transitive subgroups $G\leq \mathrm{Hol}(N)$ guiding the analysis. However, the paper also identifies rare instances where a parallel extension admits no Hopf–Galois structure of any type, providing explicit examples (notably for degree $8$) and discussing an approach to generate infinite families, while conjecturing no squarefree-degree examples exist. Overall, the results illuminate the transfer of Hopf–Galois properties along parallel subextensions and significantly constrain the behavior in the degree-$pq$ regime.

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.

Figures (2)

• Figure 1: A parallel extension, $L'/K$, to $L/K$.
• Figure 2: Example showing a Galois parallel extension.

Theorems & Definitions (39)

• Remark 2.1
• Theorem 2.2: Hall, 1928. See 9.1.7 of Rob96
• Remark 2.3
• Lemma 2.4
• proof
• Corollary 2.5
• proof
• Proposition 2.6
• proof
• Corollary 2.7