On a Question of Craven and a Theorem of Belyi
Abstract
Description
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.
Main theorem was generalized to include reducible polynomials. To appear in Proceedings of AMS
Main theorem was generalized to include reducible polynomials. To appear in Proceedings of AMS