Primitive prime divisors of \(\Phi_d(a)\) in rings of integers
Let \(L\mid\mathbb{Q}\) be an algebraic extension, \(R\) the ring of integers in \(L\). The authors ask for conditions on \(n\in \mathbb{N}\) dividing \(\Phi_d(a^n)\), \(a\in R\), \(\Phi_d\) the \(d\)-th cyclotomic polynomial. This question was solved by \textit{A. Bartholomé} [Arch. Math. 63, No. 6, 500-508 (1994; Zbl 0844.11022)] for \(R=\mathbb{Z}\). The authors now give a complete (and analogous) answer for \(L\) a quadratic extension, all prime divisors of \(n\) unramified. The problem of prime factorization of \(\Phi_d (a)\) in algebraic number fields was studied in large by \textit{H. Sachs} [Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg 6, 223-260 (1957; Zbl 0079.26802)]. Perhaps some of his results would be helpful for further investigations.
scientific article; zbMATH DE number 1145744