Divisibility of the resultant of a polynomial and a cyclotomic polynomial (Q446273)

Let \(f(x)=a\prod_{i=1}^d (x-\alpha_i)\) be a polynomial with integer coefficients and with \(a\) an integer. Note that then also \(f_l(x):=a^l\prod_{i=1}^d (x-\alpha_i^l)\) has integer coefficients. We have that \(f_l(1)\) is \(\pm\) the resultant of \(f(x)\) and \(x^l-1\). The author shows that for \(l=s^k\) the resultant is divisible by \(s^{k+1}\) if \(s|f(1)\) and establishes some related results. As a consequence he obtains a different proof of a result of \textit{M. Newman} [Ill. J. Math. 24, 156--158 (1980; Zbl 0414.15007)] on the divisibility of the determinant of an integral circular matrix.