# polynomial

## Quartic rationally derived polynomials ★★★

Author(s): Buchholz; MacDougall

Call a polynomial $p \in {\mathbb Q}[x]$ \emph{rationally derived} if all roots of $p$ and the nonzero derivatives of $p$ are rational.

\begin{conjecture} There does not exist a quartic rationally derived polynomial $p \in {\mathbb Q}[x]$ with four distinct roots. \end{conjecture}

Keywords: derivative; diophantine; elliptic; polynomial