Some results on radical extensions
type: Item
instance of: scholarly article
title: Some results on radical extensions
zbMATH Open document ID: 0798.12005
author: William Yslas Vélez
author: Fernando Barrera Mora
published in: Journal of Algebra
publication date: 1994-10-31T00:00:00Z
review text: Let \(F\) be a field, \(n\) a positive integer, and \(\zeta_ n\) a primitive \(n\)th root of unity. This paper is concerned with determining conditions on a field extension \(K/F\) which make it a radical extension. All field extensions are assumed to be separable. If \(K\) is a field, then \(K^*\) denotes the multiplicative group of nonzero elements of \(K\). If \(\Omega\) is a field, \(\mu(\Omega)\) denotes the group of roots of unity in \(\Omega\). Given a group \(G\) and a \(G\)-module \(A\), \(Z^ 1(G,A)\) denotes the group of 1-cocycles with coefficients in \(A\). The group of coboundaries is denoted by \(B^ 1(G,A)\) and the first cohomology group of \(G\) with coefficients in \(A\) is defined by \(H^ 1(G,A)= Z^ 1(G,A)/B^ 1(G,A)\). The author proves the following results. Let \(\Omega/F\) be a finite Galois extension and \(G=\text{Gal} (\Omega/F)\). Let \(H\leq G\) and \(K\) be the fixed field of \(H\). Then \(K/F\) is a radical extension if and only if there exists \(\chi\in Z^ 1(G, \mu(\Omega))\) with \(H= \text{Ker } \chi\), where \(\text{Ker } \chi= \{\sigma\in G\): \(\chi(\sigma)=1\}\). Let \(\Omega/F\) be a Galois extension and assume that there exists a finite extension \(L/F\) satisfying the properties that \(L\cap \Omega= F\) and \(\mu(L\Omega)= \mu(\Omega)\). Then there is a bijection between the radical extensions of \(F\) contained in \(\Omega\) and the radical extensions of \(L\) contained in \(L\Omega\). This bijection preserves the degree of the extensions. Let \(K/F\) be of degree \(n\) with \(\Omega\) the Galois closure over \(F\). Suppose that \(\text{char }F\) does not divide \(n\) and there exists a finite extension \(L/F\) such that \(\Omega(\zeta_ n)\cap L=F\) and \(LK= L(\alpha^{1/n})\) for some \(\alpha\in L\). Then \(K= F(a^{1/n})\) for some \(a\in F\). Let \(K/F\) be an extension of degree \(n\). Assume that \(K\cap F(\zeta_ n) =F\) and \(K(\zeta_ n)/ F(\zeta_ n)\) is cyclic. Then \(K(\zeta_ n)/F\) is normal and moreover, \(K(\zeta_ n)…
reviewed by: John N. Mordeson
Mathematics Subject Classification ID: 12F99
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 554223
zbMATH Keywords: field extension
zbMATH Keywords: radical extension
zbMATH Keywords: finite Galois extension
MaRDI profile type: Publication
full work available at URL: jabr.1993.1254
OpenAlex ID: W2096269002
DOI: 10.1006/JABR.1993.1254
description: scientific article; zbMATH DE number 554223