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 not equal to 2 we also determine which of these Hopf algebras are isomorphic as F-algebras and explicitly compute their Wedderburn-Artin decompositions.

The first named author acknowledges funding support from the Faculty of Natural Sciences at Keele University. We are grateful to Prof. Alan Koch for his comments on an early draft of this paper, and to the anonymous referee for improvements to the exposition and interpretation of our results.