We study the number of real critical points of a cyclotomic polynomial $\Phi_{ n}(x)$, that is, the real roots of $\Phi_{ n}^{ \prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes, say $p_{ 1}<\cdots<p_{ k}$. We show that if the primes are "sufficiently separated" then there are exactly $2^{ k}-1$ real roots of $\Phi_{ n}^{ \prime}(x)$ and each of them is simple.
Categories: math.NT