Non-isomorphic Hopf-Galois structures with isomorphic underlying Hopf algebras

TL;DR

This paper constructs a specific degree 8 separable extension demonstrating the existence of two distinct Hopf-Galois structures sharing the same underlying Hopf algebra, highlighting the diversity of such structures.

Contribution

It provides a concrete example of a degree 8 extension with non-isomorphic Hopf-Galois structures based on isomorphic Hopf algebras, a novel explicit construction.

Findings

Existence of two non-isomorphic Hopf-Galois structures with isomorphic underlying Hopf algebras

Explicit example in degree 8 separable extension

Insight into the structure of Hopf-Galois extensions

Abstract

We give a degree 8 separable extension having two non-isomorphic Hopf-Galois structures with isomorphic underlying Hopf algebras.