On the cyclotomic polynomial
type: Item
instance of: scholarly article
title: On the cyclotomic polynomial
zbMATH Open document ID: 0512.10006
DOI: 10.1016/0022-314X(83)90017-3
published in: Journal of Number Theory
publication date: 1983-01-01T00:00:00Z
review text: This paper is divided into two parts. In the first part some of the author's previous results [J. Number Theory 16, 420--429 (1983; Zbl 0515.12004)] are generalized. E.g., if \(2<m\in\mathbb Z\), \(1 <n\in\mathbb Z\), \(\Phi_m\) the \(m\)-th cyclotomic polynomial, \(\Phi_m(x)\equiv 0\pmod n\) for some \(x\in\mathbb Z\), then there are integers \(a, b\) with \(n=a^2-m^*b^2\). This is proven under some restrictions on \(n\), and the right definition of \(m^*\) in relation to \(m\), when the class number of \(\mathbb Q(\sqrt{m^*})\) equals one. There are counter-examples that these restrictions are necessary. As application this theorem gives some well-known results of Fermat and the following: If \(p\) is an odd prime then \(x^2 - py^2\) represents all primes \(\equiv -1\pmod{4p}\). It is well-known that the only coefficients of \(\Phi_{pq}(X)\) \((p, q\) distinct primes) are \(0, 1, -1\). In the second part the author determines explicitly the right sign for each power of \(X\).
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 3807691
zbMATH Keywords: cyclotomic polynomial
zbMATH Keywords: class number
zbMATH Keywords: representation of integers by binary quadratic forms
zbMATH Keywords: norm
author: Richard A. Mollin
reviewed by: Bernd Richter
MaRDI profile type: Publication
cites work: The Midterm Coefficient of the Cyclotomic Polynomial F pq (x)
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full work available at URL: 0022-314x(83)90017-3
OpenAlex ID: W2108120810
description: scientific article; zbMATH DE number 3807691