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

DOI:
https://doi.org/10.1090/S0002-9947-00-02271-6

Published electronically:
February 28, 2000

MathSciNet review:
1491853

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-00-02271-6

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