# Enumerating Dihedral Hopf-Galois Structures Acting on Dihedral Extensions

## TL;DR

This paper explicitly enumerates dihedral Hopf-Galois structures on dihedral extensions by analyzing regular subgroups normalized by the dihedral group within permutation groups.

## Contribution

It provides a detailed enumeration of all dihedral Hopf-Galois structures acting on dihedral extensions, extending prior theoretical frameworks with explicit classifications.

## Key findings

- Explicit enumeration of dihedral Hopf-Galois structures
- Identification of normal block systems in dihedral groups
- Classification of regular subgroups normalized by dihedral groups

## 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 $\lambda(G)\leq B$. 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.