Galois extensions of prime degree
type: Item
instance of: scholarly article
title: Galois extensions of prime degree
zbMATH Open document ID: 0706.13008
DOI: 10.1016/0021-8693(89)90137-3
author: Rick Miranda
author: Cornelius Greither
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Over a commutative ring R, the isomorphism classes of \({\mathbb{Z}}/p{\mathbb{Z}}\)-Galois extensions form an abelian group, \(T_ p(R)\), the Harrison group. In this paper, the authors calculate \(T_ p(R)\) for all R in which p is invertible, but which need not contain a primitive \(p-th\) root of 1. The method is to consider \(S=R[X]/(X^{p-1}+\cdots+X+1)\), apply Kummer theory to S to obtain a short exact sequence \(1\to S^{\times}/(S^{\times})^ p\to T_ p(S)\to Pic(S)[p]\to 1,\) apply the functor \((\;)^{\Gamma}\) to this sequence, where \(\Gamma \cong ({\mathbb{Z}}/p{\mathbb{Z}})^*\), and then show that \(T_ p(R)\cong T_ p(S)^{\Gamma}\).
Mathematics Subject Classification ID: 13B05
zbMATH DE Number: 4158782
zbMATH Keywords: Galois descent
zbMATH Keywords: Galois extensions
zbMATH Keywords: Harrison group
zbMATH Keywords: Kummer theory
reviewed by: Margaret Beattie
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90137-3
OpenAlex ID: W2053507577
cites work: Q5518494
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description: scientific article; zbMATH DE number 4158782