Primitive prime divisors of \(\Phi_d(a)\) in rings of integers
type: Item
instance of: scholarly article
title: Primitive prime divisors of \(\Phi_d(a)\) in rings of integers
zbMATH Open document ID: 0909.11047
author: Andreas Bartholomé
author: Ulrich Porsch
published in: Archiv der Mathematik
publication date: 1999-04-08T00:00:00Z
review text: Let \(L\mid\mathbb{Q}\) be an algebraic extension, \(R\) the ring of integers in \(L\). The authors ask for conditions on \(n\in \mathbb{N}\) dividing \(\Phi_d(a^n)\), \(a\in R\), \(\Phi_d\) the \(d\)-th cyclotomic polynomial. This question was solved by \textit{A. Bartholomé} [Arch. Math. 63, No. 6, 500-508 (1994; Zbl 0844.11022)] for \(R=\mathbb{Z}\). The authors now give a complete (and analogous) answer for \(L\) a quadratic extension, all prime divisors of \(n\) unramified. The problem of prime factorization of \(\Phi_d (a)\) in algebraic number fields was studied in large by \textit{H. Sachs} [Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg 6, 223-260 (1957; Zbl 0079.26802)]. Perhaps some of his results would be helpful for further investigations.
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 1145744
zbMATH Keywords: cyclotomic polynomial
zbMATH Keywords: prime factorization
zbMATH Keywords: algebraic number fields
reviewed by: Bernd Richter
MaRDI profile type: Publication
full work available at URL: s000130050111
OpenAlex ID: W2331698531
DOI: 10.1007/S000130050111
description: scientific article; zbMATH DE number 1145744