Resultants of cyclotomic polynomials (Q5894767)

Let \(\Phi_k(x)\) denote the \(k\)th cyclotomic polynomial. Let \(f\) and \(g\) be polynomials of degree \(m\), respectively \(n\) with leading coefficient \(f_m\), respectively \(g_n\). The resultant of \(f\) and \(g\) is defined as \(f_m^n g_n^m\prod_{i,j}(\alpha_i-\beta_j),\) where \(\alpha_i, \beta_j\) runs over the roots of \(f\), respectively \(g\). The author reproves two well-known results. Let \(m\) and \(n\) be positive integers with \(m<n\). The first result states that there are polynomials \(u(x),v(x)\) with integer coefficients for which \(\Phi_m(x)u(x)+\Phi_n(x)v(x)=k\), where \(k=p\) if \(n=mp^t\) for some prime \(p\) and positive integer \(t\) and \(k=1\) otherwise; he also shows that \(k\) is the smallest positive integer so represented. The author provides explicit formulas for \(u(x)\) and \(v(x)\), something which has not been done before. This result is then used to evaluate the resultant of \(\Phi_m(x)\) and \(\Phi_n(x)\). The proof of the first result uses only elementary facts about cyclotomic polynomials, the proof of the second in addition some basic statements about determinants and resultants. Earlier \textit{M. Filaseta} [Number theory for the millennium II. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, 2000. Natick, MA: A K Peters, 1--24 (2002; Zbl 1029.11007)] used the evaluation of the resultant of \(\Phi_m(x)\) and \(\Phi_n(x)\) to prove the first result. It follows that the first and second result are actually equivalent.