Classification of the Hopf Galois structures on prime power radical extensions
type: Item
instance of: scholarly article
title: Classification of the Hopf Galois structures on prime power radical extensions
zbMATH Open document ID: 0953.12003
author: Timothy Kohl
published in: Journal of Algebra
publication date: 1998-10-21T00:00:00Z
review text: Let \(k\) be a field of characteristic zero and \(K= k(w)\) a field extension generated by an element \(w\) with \(w^{p^n}= a\in k\), \(p\) a prime. Furthermore assume that \([K:k]= p^n\). Such an extension is finite and separable, but not (in general) Galois. It is, however, Hopf Galois with a Hopf algebra \(H\) which is a \(\widetilde{K}\)-Hopf algebra form of a group ring \(kN\), where \(\widetilde{K}\) is the normal closure of \(K/k\). The paper classifies the different structures of Hopf Galois extensions, in particular the possible Hopf algebras \(H\), depending on the largest integer \(r\) such that \(K\) contains a primitive \(p^r\)-th root of unity. If \(r< n\), the author shows that there are \(p^r\) Hopf Galois structures on \(K/k\).
reviewed by: Bodo Pareigis
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 1213805
zbMATH Keywords: Hopf Galois extensions
zbMATH Keywords: Hopf algebras
MaRDI profile type: Publication
full work available at URL: jabr.1998.7479
OpenAlex ID: W2006566308
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DOI: 10.1006/JABR.1998.7479
description: scientific article; zbMATH DE number 1213805