  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 $D_p$, $p\ge 3$ 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.
