On a Question of Craven and a Theorem of Belyi

Abstract

In this elementary note we prove that a polynomial with rational coefficients divides the derivative of some polynomial which splits in $\Q$ if and only if all of its irrational roots are real and simple. This provides an answer to a question posed by Thomas Craven. Similar ideas also lead to a variation of the proof of Belyi's theorem that every algebraic curve defined over an algebraic number field admits a map to $P^1$ which is only ramified above three points. As it turned out, this variation was noticed previously by G. Belyi himself and Leonardo Zapponi.