Cyclotomic polynomials at roots of unity

Abstract

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 primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for ${\Phi }_n(x)$ at the other roots of unity. This allows one to explicitly evaluate ${\Phi }_n(e^{2\pi i/m})$ 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 ${\Phi }_n(x)$. We also obtain a formula for ${\Phi }_n^{\prime }(e^{2\pi i/m})/{\Phi }_n(e^{2\pi i/m})$ with $n\ne m$, which is effectively applied to $m\in \{3,4,6\}$. Furthermore, we compute the resultant of two cyclotomic polynomials in a novel very short way.

17 pages, 4 tables, substantially reworked version. New and and very short computation of the resultant of two cyclotomic polynomials added, also results for m=5,8,10 and 12. Discussion of numerical semigroups, Coxeter polynomials and the incorrect Proposition 14 are left oUT