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

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.

15 pages, some arguments shortened, some more exposition included