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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Bivariate factorizations connecting Dickson polynomials and Galois theory


Authors: Shreeram S. Abhyankar, Stephen D. Cohen and Michael E. Zieve
Journal: Trans. Amer. Math. Soc. 352 (2000), 2871-2887
MSC (1991): Primary 12E05, 12F10, 14H30, 20D06, 20G40, 20E22
Published electronically: February 28, 2000
MathSciNet review: 1491853
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Abstract:

In his Ph.D. Thesis of 1897, Dickson introduced certain permutation polynomials whose Galois groups are essentially the dihedral groups. These are now called Dickson polynomials of the first kind, to distinguish them from their variations introduced by Schur in 1923, which are now called Dickson polynomials of the second kind. In the last few decades there have been extensive investigations of both of these types, which are related to the classical Chebyshev polynomials. We give new bivariate factorizations involving both types of Dickson polynomials. These factorizations demonstrate certain isomorphisms between dihedral groups and orthogonal groups, and lead to the construction of explicit equations with orthogonal groups as Galois groups.


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Additional Information

Shreeram S. Abhyankar
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: ram@cs.purdue.edu

Stephen D. Cohen
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland
Email: sdc@maths.gla.ac.uk

Michael E. Zieve
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
Email: zieve@math.brown.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02271-6
PII: S 0002-9947(00)02271-6
Received by editor(s): July 3, 1997
Received by editor(s) in revised form: November 21, 1997
Published electronically: February 28, 2000
Additional Notes: Abhyankar’s work was partly supported by NSF grant DMS 91-01424 and NSA grant MDA 904-97-1-0010. Zieve’s work was partly supported by an NSF postdoctoral fellowship. Abhyankar and Zieve were also supported by EPSRC Visiting Fellowship GR/L 43329.
Article copyright: © Copyright 2000 American Mathematical Society