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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Fractalized cyclotomic polynomials


Author: David P. Roberts
Journal: Proc. Amer. Math. Soc. 135 (2007), 1959-1967
MSC (2000): Primary 11R21; Secondary 12E10, 37F99
Published electronically: February 28, 2007
MathSciNet review: 2299467
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Abstract: For each prime power $ p^m$, we realize the classical cyclotomic polynomial $ \Phi_{p^m}(x)$ as one of a collection of $ 3^m$ different polynomials in $ \mathbf{Z}[x]$. We show that the new polynomials are similar to $ \Phi_{p^m}(x)$ in many ways, including that their discriminants all have the form $ \pm p^c$. We show also that the new polynomials are more complicated than $ \Phi_{p^m}(x)$ in other ways, including that their complex roots are generally fractal in appearance.


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Additional Information

David P. Roberts
Affiliation: Division of Science and Mathematics, University of Minnesota-Morris, Morris, Minnesota 56267
Email: roberts@morris.umn.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08629-7
PII: S 0002-9939(07)08629-7
Keywords: Cyclotomic polynomial, discriminant, fractal, Galois
Received by editor(s): November 2, 2005
Received by editor(s) in revised form: January 4, 2006
Published electronically: February 28, 2007
Communicated by: Ken Ono
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.