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 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 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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Additional Information

**Alexander Borisov**

Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Email:
borisov@math.psu.edu

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

Received by editor(s):
July 19, 2002

Published electronically:
July 2, 2003

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 2003
American Mathematical Society