Cyclotomic polynomials at roots of unity

Bartlomiej Bzdega, Andres Herrera-Poyatos, Pieter Moree

Published 2016-11-21Version 1

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\Phi_n(x)$ is trivially zero at $n^{th}$ roots of unity. We investigate the values of $\Phi_n(x)$ at the other roots of unity. Motose (2006) seems to have been the first to evaluate $\Phi_n(e^{2\pi i/m})$ with $m\in \{3,4,6\}$. We reprove and correct his work. Furthermore, we give a simple reproof of a result of Apostol (1970) on the resultant of cyclotomic polynomials and of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $\Phi_n(x)$. Also we precisely characterize the $n$ and $m$ for which $\Phi_n(\zeta_m)\in \{\pm 1\}$.