Cyclotomic polynomials over cyclotomic fields (Q2904285)

It is an exercise in Galois theory to factorize cyclotomic polynomials over cyclotomic fields.NEWLINENEWLINE With \(\Phi_n\) the \(n\)th cyclotomic polynomial, \(\zeta_n\) a primitive root of unity, the authors give the minimal polynomial of \(\zeta^k_n\) over \({\mathbb Q}(\zeta_m)\), \((k, n)= 1\). So they are ready to factorize \(\Phi_n(x)\) over \({\mathbb Q}(\zeta_m)\).NEWLINENEWLINE As an application, they are able to give recursively the coefficients of \(\Phi_{3n}(x)\) by those of \(\Phi_n(x)\) if \(3\nmid n\). By the way they determine the coefficients of \(x^{\varphi(p q)}\) of \(\Phi_{3pq}(x)\), \(p\neq q\), prime.