Critical points, critical values of a prime polynomial

Abstract

Given a polynomial f (x), we study the possibility of expressing it as the composition of two non-constant and non-linear polynomials. In this case f(x) is said to be composite otherwise it is prime. We give sufficient conditions for a polynomial to be prime in terms of its critical values and critical points. Given two polynomials, f (x) and h(x) we give methods to decide if h(x) is a right composition factor of f (x) and in that case to find the polynomial g(x) such that f = g ○ h. Finally we propose an algorithm to decompose a polynomial f (x) into its prime factors if one knows its list of critical points with their valencies.