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

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

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.

**1.**C. Ballantine, J. Roberts. A simple proof of Rolle's theorem for finite fields,*Amer. Math. Monthly***109**(2002), no. 1, 72-74. MR**2003b:12003****2.**G. V. Belyi. Galois extensions of a maximal cyclotomic field. (Russian)*Izv. Akad. Nauk SSSR Ser. Mat.***43**(1979), no. 2, 267-276. MR**80f:12008****3.**G. V. Belyi, Another Proof of Three Points Theorem, preprint, 1997. http://www.mpim-bonn.mpg.de/html/preprints/preprints.html**4.**Thomas C. Craven. A weak version of Rolle's theorem.*Proc. Amer. Math. Soc.***125**(1997), no. 11, 3147-3153. MR**97m:12010****5.**Irving Kaplansky. Fields and rings. Second edition.*Chicago Lectures in Mathematics*. The University of Chicago Press, Chicago, Ill.-London, 1972. MR**50:2139****6.**Jürgen Wolfart. Kinderzeichnungen und Uniformisierungstheorie, preprint, 2001. http://www. math.uni-frankfurt.de/ steuding/wolfart.shtml**7.**Leonardo Zapponi. A simplification in the proof of Belyi's theorem, preprint.

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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