Hopf Forms and Hopf-Galois Theory

TL;DR

This paper explores the relationship between $F$-Galois extensions and Hopf forms of group rings over $ ext{Q}$, providing explicit constructions and classifications for specific groups and conditions.

Contribution

It establishes a bijection between $F$-Galois extensions and Hopf forms of group rings, and explicitly constructs Hopf forms for certain finite groups.

Findings

Existence of absolutely semisimple Hopf forms for specific groups

Explicit construction of Galois extensions corresponding to these Hopf forms

Method to recover Galois extensions from Hopf forms under certain conditions

Abstract

Let $K$ be a finite field extension of $\Q$ and let $N$ be a finite group with automorphism group $F=\Aut(N)$. R. Haggenm\"{u}ller and B. Pareigis have shown that there is a bijection \[\Theta: {\mathcal Gal}(K,F)\rightarrow {\mathcal Hopf}(K[N])\] from the collection of $F$-Galois extensions of $K$ to the collection of Hopf forms of the group ring $K[N]$. For $N=C_n$, $n\ge 1$, $C_p^m$, $p$ prime, $m\ge 1$, and $N=D_3,D_4,Q_8$, we show that $\Q[N]$ admits an absolutely semisimple Hopf form $H$ and find $L$ for which $\Theta (L)=H$. Moreover, if $H$ is the Hopf algebra given by a Hopf-Galois structure on a Galois extension $E/K$, we show how to construct the preimage of $H$ under $\Theta$ assuming certain conditions.