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 $N\leq B=Perm(G)$ is a regular subgroup that is normalized by the left regular representation $λ(G)\leq B$. We consider the case where both $G$ and $N$ are isomorphic to a dihedral group $D_n$ for any $n\geq 3$. Using the normal block systems inherent to the left regular representation of each $D_n$,(and every other regular permutation group isomorphic to $D_n$) we explicitly enumerate all possible such $N$ which arise.