Cyclotomic polynomials at roots of unity

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