A note on cyclotomic polynomials
type: Item
instance of: scholarly article
title: A note on cyclotomic polynomials
zbMATH Open document ID: 0989.11013
author: Ming-Chang Kang
published in: Rocky Mountain Journal of Mathematics
publication date: 2002-07-24T00:00:00Z
full work available at URL: CONT29-3.html
review text: After recalling an identity concerning cyclotomic polynomials found by \textit{C. C. Cheng, J. H. McKay} and \textit{S. S. Wang} [Proc. Am. Math. Soc. 123, 1053-1059 (1995; Zbl 0828.11014)], the author gives three applications of it. First a description is given of polynomials \(f\in\mathbb{Z}[X]\) satisfying \(f(X)|f(X^n)\) for some fixed \(n\geq 2\). Then a characterization is obtained for sequences \(f_1,f_2,\dots\) of polynomials with integral coefficients, for which the identities \(f_m(X^n)=\prod_{d|n}f_{md}(X)\) hold for all coprime integers \(m,n\). Finally a sufficient condition is provided for a finite extension \(K\) of the rationals to have the property that the quotient ring \(Z_K/(X^n-1)Z_K\) (with \(Z_K\) being the ring of integers of \(K\)) is a Dedekind-like ring, as defined by \textit{L. S. Levy} [J. Algebra 93, 1-116 (1985; Zbl 0564.13010)].
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 12E10
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 20C10
zbMATH DE Number: 1439641
zbMATH Keywords: cyclotomic polynomials
zbMATH Keywords: integral group rings
zbMATH Keywords: polynomials with integral coefficients
zbMATH Keywords: finite extension of rationals
zbMATH Keywords: Dedekind-like ring
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
OpenAlex ID: W2014734635
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DOI: 10.1216/RMJM/1181071615
description: scientific article; zbMATH DE number 1439641