Cyclotomic polynomials at roots of unity

The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $Φ_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $Φ_n(x)$ at the other roots of unity. This allows one to explicitly evaluate $Φ_n(e^{2πi/m})$ 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 $Φ_n(x)$. We also obtain a formula for $Φ_n'(e^{2πi/m}) / Φ_n(e^{2πi/m})$ 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.

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