Hopf-Galois module structure of a class of tame quaternionic fields

Abstract

We study the Hopf-Galois module structure of rings of integers in tame Galois extensions L=F of global fields with Galois group isomorphic to the quaternion group of order 8. We determine explicitly the Hopf algebras giving Hopf-Galois structures on such extensions and study which of these are isomorphic as Hopf algebras or as F-algebras. We study "quotient" structures in order to understand the Hopf-Galois module structure in such extensions corresponding to Hopf algebras of cyclic type.
Next we specialise to a certain family of tame quaternionic extensions, L/?, employing a construction of Fujisaki. We show that for these extensions the ring of algebraic integers, ?L, is locally free over its associated order in each of the Hopf-Galois structures. We find explicit local generators for all but the structures of cyclic type. We then employ the machinery of locally free class groups to study the structures of dihedral type and give necessary and sufficient conditions for ?L to be free over its associated order in each of these structures.