KöMaL 2010 A517

Let $m \ge$3 be a positive integer, and let $\Phi_m$($x$) be the $m$th cyclotomic polynomial, and denote by $\Psi_m$($x$) the polynomial with integer coefficients for which $x^{\varphi(m)/2} \Psi_m\left(x+\frac1x\right) = \Phi_m(x)$. Prove that for every integer $a$, any prime divisor of the number $\Psi_m$($a$) either divides $m$ or is of the form $mk \pm$1.