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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.'
author:
- 'Stuart Taylor, Paul J. Truman'
title: 'The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions'
---

Introduction
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Let $L/F$ be a finite Galois extension of fields with group $G$. The group algebra $ F[G] $, with its usual action on $ L $, is an example of a [*Hopf-Galois structure*]{} on the extension. If $H$ is a finite dimensional $F$-Hopf algebra, then we say that $H$ gives a Hopf Galois structure on $L/F$ if and only if the following conditions hold:

$L$ is an $H$-module algebra; that is: $L$ is an $H$-module with action $h(x)$ for $h\in H$ and $x\in L$ where, for $y\in L$, $h(xy)=\sum_{(h)}h_{(1)}(x)h_{(2)}(y)$ (Sweedler notation) and $h(1)=\epsilon(h)(1)$;\

the $F$-linear map $j:L\otimes_F H\to{\text{End}}_K(L)$ given 