The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions

TL;DR

This paper classifies all Hopf-Galois structures on quaternionic Galois extensions, identifies isomorphisms among the resulting Hopf algebras, and computes their algebraic decompositions in characteristic zero.

Contribution

It provides a complete description of Hopf-Galois structures for quaternionic extensions and analyzes their algebraic isomorphisms and decompositions.

Findings

Classification of all Hopf-Galois structures on quaternionic extensions

Identification of isomorphic Hopf algebras among these structures

Explicit Wedderburn-Artin decompositions in characteristic zero

Abstract

Let $L/F$ be a Galois extension of fields with Galois group isomorphic to the quaternion group of order $ 8 $. We describe all of the Hopf-Galois structures admitted by $ L/F $, and determine which of the Hopf algebras that appear are isomorphic as Hopf algebras. In the case that $ F $ has characteristic zero we also determine which of these Hopf algebras are isomorphic as $ F $-algebras and explicitly compute their Wedderburn-Artin decompositions.