A discriminant criteria for reducibility of a polynomial
The authors consider sufficient conditions for a polyomial \(p(w)\) over a domain \(K\) (a commutative ring without zero divisors) with unity to split into linear factors in \(K\), and intend to generalize the Motzkin-Taussky theorem to the case \(K={\mathbb{C}}[z]\) or \({\mathbb{Z}}\) [cf. \textit{T. S. Motzkin} and \textit{O. Taussky}, Trans. Am. Math. Soc. 80, 387-401 (1955; Zbl 0067.25401)]. For this purpose, they use the Riemann-Hurwitz formula (in case of \(K={\mathbb{C}}[z])\) and a theorem of Minkowski (in case of \(K={\mathbb{Z}})\), and obtain an additional condition on the roots of \(p(w)\) which together with the discriminant condition implies the splitting of \(p(w)\).
scientific article; zbMATH DE number 3968704