Prime factorization for values of cyclotomic polynomials in \({\mathbb Z}[i]\)
Let \(\Phi _ n(x)\) be the n-th cyclotomic polynomial. The values \(\Phi _ n(e)\) for given \(e\in {\mathbb{Z}}[i]\) have the property that their prime divisors are characterized by certain congruence conditions. This is a generalization of a known theorem of Sylvester, Kronecker and Zsigmondy on the prime factorization of \(\Phi _ n(e)\) with given \(e\in {\mathbb{Z}}\). Our result improves the corresponding results of \textit{L. Rédei} [Acta Sci. Math. 19, 98-126 (1958; Zbl 0085.025)] and \textit{H. Sachs} [Wiss. Z., Martin-Luther-Univ. Halle-Wittenberg 6, 223-260 (1957; Zbl 0079.268)]. Further we present new estimates of \(| \Phi _ n(x)|\) for \(x\in {\mathbb{C}}\) \((| x| >1)\).
scientific article; zbMATH DE number 3987390