The Structure of Hopf Algebras Acting on Galois Extensions with Dihedral Groups

Alan Koch, Timothy Kohl, Paul J. Truman, Robert Underwood

Published 2017-08-31Version 1

Let $L/K$ be a finite separable extension of fields. The Hopf Galois structures on $L/K$ have been classified by C. Greither and B. Pareigis. In the case that $L/K$ is finite separable and Galois with non-abelian group $G$, $L/K$ admits both a classical Hopf Galois structure via the $K$-Hopf algebra $KG$ and a canonical non-classical structure via the $K$-Hopf algebra $H$. In this paper we investigate the structure of $H$ as a $K$-Hopf algebra and as a $K$-algebra. For $G$ non-abelian, we show that $KG\not \cong H$ as $K$-Hopf algebras, and we prove that $KG$ and $H$ have the same number of components in their Wedderburn-Artin decompositions. We specialize to the case where $K=\mathbb Q$, and $L/\mathbb Q$ is Galois with group $D_n$, the dihedral group of order $2n$, $n\ge 3$. For $n=3,4$, we give necessary and sufficient conditions on $L/\mathbb Q$ under which $\mathbb Q D_n\cong H$ as $K$-algebras. We include some results in the case $n$ is prime, with particular attention to $n=5$.