On a theorem of Rédei on cyclotomic polynomials.
type: Item
instance of: scholarly article
title: On a theorem of Rédei on cyclotomic polynomials.
zbMATH Open document ID: 0655.10009
DOI: 10.1007/BF01952473
author: S. H. Smith
published in: Acta Mathematica Hungarica
publication date: 1988-01-01T00:00:00Z
review text: Let n be a natural number, \(\zeta\) a primitive nth root of unity, \(C_ n=\) the cyclic group with n elements. A theorem of \textit{I. J. Schoenberg} [Mathematika 11, 131-136 (1964; Zbl 0132.256)] gives all relations \(\sum^{n}_{k=1}c_ k\zeta^ k=0\), \(c_ k\in {\mathbb{Z}}\). This is: Let \(\phi\) : \({\mathbb{Z}}[C_ n]\to {\mathbb{Z}}[\zeta]\) be the epimorphism from the group ring \({\mathbb{Z}}[C_ n]\) onto the nth cyclotomic integers given by \(a\mapsto \zeta\). Let \(p| n\) be a prime and \(a_{p,k}=a^ k\sum^{p}_{\lambda =1}a^{\lambda n/p}\in {\mathbb{Z}}[C_ n]\). Clearly \(a_{p,k}\in Kern \phi\). The theorem of Schoenberg says that Kern \(\phi\) is \({\mathbb{Z}}\)-linearly generated by all these \(a_{p,k}\) (p\(| n\), \(k=1,...,n)\). The author gives a direct algebraic proof of this and derives the following theorem of Rédei: Let \(\Phi_ n(X)\) be the nth cyclotomic polynomial. For all \(p| n\) prime there are integral polynomials \(f_{p,n}(X)\) such that \[ \Phi_ n(X)=\sum_{p| n}f_{p,n}(X) \Phi_ p(X^{n/p}). \] Apparently not knowing this theorem of \textit{L. Rédei} [Acta Math. Acad. Sci. Hung. 1, 197-206 (1950; Zbl 0041.15703); ibid. 5, 27-28 (1954; Zbl 0055.013)], Schoenberg gave a complicated proof by induction for this formula to obtain his theorem, a way the author successfully avoids.
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11T22
zbMATH DE Number: 4068969
zbMATH Keywords: theorem of Schoenberg
zbMATH Keywords: theorem of Rédei
zbMATH Keywords: cyclotomic polynomial
reviewed by: Bernd Richter
MaRDI profile type: Publication
cites work: Q5820591
cites work: Ein Beitrag zum Problem der Faktorisation von endlichen Abelschen Gruppen
cites work: Über das Kreisteilungspolynom
cites work: A note on the cyclotomic polynomial
description: scientific article; zbMATH DE number 4068969