A note on cyclotomic polynomials
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)].
scientific article; zbMATH DE number 1439641
scientific article