The Structure of Hopf Algebras Acting on Dihedral Extensions

We discuss isomorphism questions concerning the Hopf algebras that yield Hopf–Galois structures for a fixed separable field extension L/K. We study in detail the case where L/K is Galois with dihedral group Dp, p≥3 prime and give explicit descriptions of the Hopf algebras which act on L/K. We also determine when two such Hopf algebras are isomorphic, either as Hopf algebras or as algebras. For the case p=3 and a chosen L/K, we give the Wedderburn–Artin decompositions of the Hopf algebras