On ${\mathbb{Z}}_N⋊{\mathbb{Z}}_2$-Hopf-Galois structures

Abstract

Let $K/F$ be a finite Galois extension of fields with $Gal(K/F)=\Gamma$. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. Dihedral group is one particular example of semidirect product of ${\mathbb{Z}}_n$ and ${\mathbb{Z}}_2$. In this article we count the number of Hopf-Galois structures with Galois group $\Gamma$ of type $G$, where $\Gamma ,G$ are groups of the form ${\mathbb{Z}}_N{⋊}_{\varphi }{\mathbb{Z}}_2$ when $N$ is odd with radical of $N$ being a Burnside number. As an application we also find the corresponding number of skew braces.