Hopf--Galois structures of cyclic type on parallel extensions of prime power degree

TL;DR

This work addresses whether parallel extensions preserve cyclic Hopf--Galois structures of type $N$ for a given base extension $L/K$, and provides a complete resolution when $[L:K]$ is an odd or even prime power with $N$ cyclic. The authors employ a group-theoretic framework via Greither--Pareigis and Byott, recasting Hopf--Galois structures as transitive subgroups of the holomorph $ ext{Hol}(N)$ and analyzing stabiliser-conjugacy data. They show that for odd prime-power degrees, any parallel extension carrying a cyclic $N$-type Hopf--Galois structure must be conjugate to $L/K$, while in the even prime-power case a dichotomy emerges: if $G$ has an element of order $[L:K]$, all parallel extensions admit the cyclic structure; otherwise, some parallel extensions do not, with a complete group-theoretic classification of obstructed cases. The results illuminate the behavior of cyclic-type Hopf--Galois structures under the parallel-extension operation and provide precise criteria for when such structures persist or fail.

Abstract

Let $L/K$ be any finite separable extension with normal closure $\widetilde{L}/K$. An extension $L'/K$ is said to be $\textit{parallel to $L/K$}$ if $L'$ is an intermediate field of $\widetilde{L}/K$ with $[L':K]=[L:K]$. We study the following question -- Given that $L/K$ admits a Hopf--Galois structure of type $N$, does it imply that every extension parallel to $L/K$ also admits a Hopf--Galois structure of type $N$? We completely solve this problem when the degree $[L:K]$ is a prime power and the type $N$ is cyclic. Our approach is group-theoretic and uses the work of Greither--Pareigis and Byott.

Theorems & Definitions (45)

• Theorem 1.4
• proof
• Theorem 1.5
• proof
• Theorem 1.6
• proof
• Remark 1.7
• Remark 1.8
• Lemma 2.1
• proof