cyclotomic polynomials and units in cyclotomic number fields
type: Item
instance of: scholarly article
title: cyclotomic polynomials and units in cyclotomic number fields
zbMATH Open document ID: 0634.12005
DOI: 10.1016/0022-314X(88)90043-1
author: Michael Kaminski
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: The author proves (theorem 1) that if P(x)\(\neq x\) is a monic irreducible polynomial with integer coefficients such that its resultant with infinitely many cyclotomic polynomials is \(+1\) or -1, then P(x) is a cyclotomic polynomial. From this he deduces a number of interesting corollaries: for example, if \(\alpha\neq 0\) is an algebraic integer such that for infinitely many n, 1- \(\alpha\) n is a unit in the ring of algebraic integers \(Z[\alpha]\), then \(\alpha\) is a root of unity. The proof uses a theorem of \textit{A. Baker} on linear forms in the logarithms of algebraic numbers [see Acta Arith. 21, 117-129 (1972; Zbl 0244.10031)]. Thus the finite number of exceptions in theorem 1 and its corollaries can be effectively determined for a given noncyclotomic P(x).
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 4031818
zbMATH Keywords: resultant
zbMATH Keywords: cyclotomic polynomial
zbMATH Keywords: root of unity
zbMATH Keywords: logarithms of algebraic numbers
reviewed by: David W. Boyd
MaRDI profile type: Publication
cites work: Resultants of Cyclotomic Polynomials
cites work: A sharpening of the bounds for linear forms in logarithms
cites work: Q5588717
cites work: Q5612629
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description: scientific article; zbMATH DE number 4031818