Quartic rationally derived polynomials
Call a polynomial rationally derived if all roots of and the nonzero derivatives of are rational.
Probably anyone who has ever designed simple problems for calculus students has looked for polynomials with the property that both and some small derivatives of it are easy to factor. Perhaps inspired by this, Buchholz and MacDougall attempted to classify all univariate polynomials defined over a domain with the property that they and all their nonzero derivatives have all their roots in . This problem can be split into cases dependent upon the multiplicity of the roots, and Buchholz and MacDougall solved many of the small ones for . Based on their results and a theorem of Flynn [F], an affirmative solution to the above conjecture would complete this classification problem for .
*[BM] R. Buchholz, and J. MacDougall, When Newton met Diophantus: a study of rational-derived polynomials and their extension to quadratic fields. J. Number Theory 81 (2000), no. 2, 210--233. MathSciNet
[F] E. V. Flynn, On Q-derived polynomials. Proc. Edinb. Math. Soc. (2) 44 (2001), no. 1, 103--110. MathSciNet
* indicates original appearance(s) of problem.