Cyclotomic polynomials at roots of unity
Volume 184 / 2018
Acta Arithmetica 184 (2018), 215-230
MSC: Primary 11C08; Secondary 11R09.
DOI: 10.4064/aa170112-20-12
Published online: 27 July 2018
Abstract
The $n$th cyclotomic polynomial $\varPhi _n(x)$ is the minimal polynomial of an $n$th primitive root of unity. Hence $\varPhi _n(x)$ is trivially zero at primitive $n$th roots of unity. Using finite Fourier analysis we derive a formula for $\varPhi _n(x)$ at the other roots of unity. This allows one to explicitly evaluate $\varPhi _n(e^{2\pi i/m})$ with $m\in \{ 3,4,5,6,8,10,12\} $. We use this evaluation with $m=5$ to give a simple reproof of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $\varPhi _n(x)$. Furthermore, we compute the resultant of two cyclotomic polynomials in a novel very short way.
