A characterization of separable polynomials over a ring
type: Item
instance of: scholarly article
title: A characterization of separable polynomials over a ring
zbMATH Open document ID: 0606.16001
DOI: 10.1007/BF01961006
author: George Szeto
author: Y.-F. Wong
published in: Acta Mathematica Hungarica
publication date: 1985-01-01T00:00:00Z
review text: Let R be a ring, not necessarily commutative, having a unity. Write \(S=R[X]\) for the polynomial ring in an indeterminate X satisfying \(rX=Xr\) for each \(r\in R\). Suppose that f(X)\(\in S\) is monic and that \(I=(f(X))\) is the ideal of S generated by f(X). Then f(X) is called a separable polynomial over R if \(\bar S=S/I\) is a separable ring extension over R having a free basis \(\{1,x,...,x^{n-1}\}\), where \(x=X+I\). Given \(s\in \bar S\), we define the trace t(s) to be \(\sum_{i}\pi_ i(sx^ i)\), where \(\pi_ i\) is the \(i^{th}\) projection map, and we let T be the \(n\times n\) matrix with \(t(x^ ix^ j)\) as the entry in position \((i+1,j+1)\), for \(i,j=0,1,...,n-1.\) \textit{F. DeMeyer} and \textit{E. Ingraham} [Separable algebras over commutative rings (Lect. Notes Math. 181, 1971; Zbl 0215.366)] have shown that if R is commutative then f(X) is separable over R if and only if the determinant of T is a unit in R. This theorem is generalized in the present paper. It is shown that (i) when R is finitely generated and projective over its centre, f(X) is separable if and only if T is invertible and (ii) when R is non-commutative, finitely generated and projective over its centre C, then f(X) is separable if and only if the determinant of T is a unit in C.
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S20
zbMATH DE Number: 3979274
zbMATH Keywords: polynomial ring
zbMATH Keywords: separable polynomial
zbMATH Keywords: separable ring extension
zbMATH Keywords: free basis
zbMATH Keywords: determinant
zbMATH Keywords: unit
MaRDI profile type: Publication
cites work: Separable algebras over commutative rings
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cites work: A characterization of a cyclic Galois extension of commutative rings
cites work: On free quaternion extensions of rings
cites work: Q3949161
full work available at URL: https://doi.org/10.1007/bf01961006
OpenAlex ID: W2072827629
description: scientific article; zbMATH DE number 3979274