Hopf–Galois structures on parallel extensions

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 $L^{\prime }/K$ of $E/K$ parallel to $L/K$ if $[L^{\prime }:K]=n$.

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.