Prime factorization for values of cyclotomic polynomials in \({\mathbb Z}[i]\)
type: Item
instance of: scholarly article
title: Prime factorization for values of cyclotomic polynomials in \({\mathbb Z}[i]\)
zbMATH Open document ID: 0611.12001
DOI: 10.1007/BF01210712
author: Gabriele Steidl
author: Manfred Tasche
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Phi _ n(x)\) be the n-th cyclotomic polynomial. The values \(\Phi _ n(e)\) for given \(e\in {\mathbb{Z}}[i]\) have the property that their prime divisors are characterized by certain congruence conditions. This is a generalization of a known theorem of Sylvester, Kronecker and Zsigmondy on the prime factorization of \(\Phi _ n(e)\) with given \(e\in {\mathbb{Z}}\). Our result improves the corresponding results of \textit{L. Rédei} [Acta Sci. Math. 19, 98-126 (1958; Zbl 0085.025)] and \textit{H. Sachs} [Wiss. Z., Martin-Luther-Univ. Halle-Wittenberg 6, 223-260 (1957; Zbl 0079.268)]. Further we present new estimates of \(| \Phi _ n(x)|\) for \(x\in {\mathbb{C}}\) \((| x| >1)\).
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11A41
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 3987390
zbMATH Keywords: Gaussian integers
zbMATH Keywords: cyclotomic polynomial
zbMATH Keywords: prime factorization
Wikidata QID: Q114234009
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3987390