Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a question of Craven and a theorem of Belyi

Author: Alexander Borisov
Journal: Proc. Amer. Math. Soc. 131 (2003), 3677-3679
MSC (2000): Primary 11R80; Secondary 11G99
Published electronically: July 2, 2003
MathSciNet review: 1998173
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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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Alexander Borisov
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Received by editor(s): July 19, 2002
Published electronically: July 2, 2003
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2003 American Mathematical Society