On a Question of Craven and a Theorem of Belyi

Details On a Question of Craven and a Theorem of Belyi On a Question of Craven and a Theorem of Belyi

2002-05-25

arxiv.org/abs/math/0205266v3

Mathematics

Number Theory

Main theorem was generalized to include reducible polynomials. To appear in Proceedings of AMS

Scientific paper

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.

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