Minimal Hopf-Galois Structures on Separable Field Extensions

TL;DR

This paper explores the special case of minimal Hopf-Galois structures on separable field extensions, focusing on situations where the Hopf algebra has only two sub-Hopf algebras, revealing insights into their group-theoretical properties.

Contribution

It provides a survey and illustration of methods to analyze Hopf-Galois structures with minimal sub-Hopf algebras on finite separable extensions.

Findings

Characterization of minimal Hopf-Galois structures with two sub-Hopf algebras

Group-theoretical methods applied to classify such structures

Insights into the relationship between sub-Hopf algebras and intermediate fields

Abstract

In Hopf-Galois theory, every $H$-Hopf-Galois structure on a field extension $K/k$ gives rise to an injective map $\mathcal{F}$ from the set of $k$-sub-Hopf algebras of $H$ into the intermediate fields of $K/k$. Recent papers on the failure of the surjectivity of $\mathcal{F}$ reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. This paper surveys and illustrates group-theoretical methods to determine $H$-Hopf-Galois structures on finite separable extensions in the extreme situation when $H$ has only two sub-Hopf algebras.