On a question of Craven and a theorem of Belyi
Author:
Alexander Borisov
Journal:
Proc. Amer. Math. Soc. 131 (2003), 36773679
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:
http://dx.doi.org/10.1090/S000299390307151X
PII:
S 00029939(03)07151X
Received by editor(s):
July 19, 2002
Published electronically:
July 2, 2003
Communicated by:
David E. Rohrlich
Article copyright:
© Copyright 2003
American Mathematical Society
