On a question of Craven and a theorem of Belyi

Borisov, Alexander

doi:10.1090/S0002-9939-03-07151-X

On a question of Craven and a theorem of Belyi
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by Alexander Borisov

Proc. Amer. Math. Soc. 131 (2003), 3677-3679

DOI: https://doi.org/10.1090/S0002-9939-03-07151-X

Published electronically: July 2, 2003

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Abstract:

In this elementary note we prove that a polynomial with rational coefficients divides the derivative of some polynomial which splits in $\mathbb Q$ 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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