The theorem of the primitive element for a semi-local ring.
type: Item
instance of: scholarly article
title: The theorem of the primitive element for a semi-local ring.
zbMATH Open document ID: 0605.13006
DOI: 10.1016/0021-8693(87)90178-5
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The main result is an extension to semi-local rings of the theorem of the primitive element for separable extensions (of finite type) of a local ring with infinite residue field. Thus it is shown that if \(A,{\mathfrak r}\) is a semilocal ring with \(A/{\mathfrak r}\) the product of fields \(k_ 1\times\dots\times k_ r\), then every separable extension of \(A\) of rank \(m\) is simple if and only if \(\text{Card}\,k_ i\geq m\) for \(i=1,\dots,r\). The note ends with some consequences for small values of \(m\).
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 3977119
zbMATH Keywords: theorem of the primitive element
zbMATH Keywords: separable extensions
author: Jean-Daniel Therond
reviewed by: David Kirby
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90178-5
OpenAlex ID: W2079068284
cites work: Q4161447
cites work: Q3963102
cites work: Q3843687
cites work: Q5518494
cites work: The Group of Unramified Kummer Extensions of Prime Degree
cites work: Separable algebras over commutative rings
cites work: Q5518497
cites work: Separable Algebras Over Commutative Rings
cites work: On the free quadratic extensions of a commutative ring
cites work: Sur le groupe des extensions cycliques
cites work: Sur une propriété des unités d'un corps algébrique
cites work: Quelques problèmes relatifs aux unités algébriques
cites work: Sur un type particulier d'unités algébriques
cites work: Q4112023
cites work: Q3936827
cites work: Existence d'une extension cyclique cubique monogene de discriminant donne
description: scientific article; zbMATH DE number 3977119