Enumerating Dihedral Hopf-Galois structures acting on Dihedral extensions

Abstract

The work of Greither and Pareigis details the enumeration of the Hopf-Galois structures (if any) on a given separable field extension. For an extension L/K which is classically Galois with G=Gal(L/K) the Hopf algebras in question are of the form (L[N])G where N≤B=Perm(G) is a regular subgroup that is normalized by the left regular representation λ(G)≤B. We consider the case where both G and N are isomorphic to a dihedral group Dn for any n≥3. Using the normal block systems inherent to the left regular representation of each Dn, (and every other regular permutation group isomorphic to Dn) we explicitly enumerate all possible such N which arise.