A Class of Profinite Hopf-Galois Extensions Over Q

TL;DR

This paper extends Hopf-Galois theory to profinite radical extensions over rationals, constructing a profinite Hopf algebra that generalizes previous results on finite radical extensions.

Contribution

It introduces a new class of profinite Hopf-Galois extensions over Q and constructs a corresponding profinite Hopf algebra acting on these extensions.

Findings

Constructed a profinite Hopf algebra for the union of radical extensions.

Extended existing Hopf-Galois classification to profinite radical extensions.

Generalized results on Hopf algebra forms of group algebras.

Abstract

For $p$ a prime and $a\in\mathbb{Q}$, where $a$ is not a $p^n$-th power of any rational number, the extension $\mathbb{Q}(w_n)/\mathbb{Q}$ where $w_n=\root p^n \of a$ is separable but non-normal. The Hopf-Galois theory for separable extensions was determined by Greither and Pareigis, and the specific classification for radical extensions such as these by the author. In this work we extend this theory to a certain class of profinite extensions, namely those formed from the union of these $\mathbb{Q}(w_n)$. We construct a 'profinite' Hopf algebra which acts, and show that it satisfies a generalization of a result due to Haggenmuller and Pareigis on the structure of Hopf algebra forms of group algebras.