When Newton met Diophantus: A Study of Rational-Derived Polynomials and Their Extension to Quadratic Fields Original Research Article

We consider the problem of classifying all univariate polynomials, defined over a domain k, with the property that they and all their derivatives have all their roots in k. This leads to a number of interesting sub-problems such as finding k-rational points on a curve of genus 1 and rational points on a curve of genus 2.