## Bivariate factorizations connecting Dickson polynomials and Galois theory

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- by Shreeram S. Abhyankar, Stephen D. Cohen and Michael E. Zieve PDF
- Trans. Amer. Math. Soc.
**352**(2000), 2871-2887 Request permission

## 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.## References

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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
- MR Author ID: 50360
- Email: sdc@maths.gla.ac.uk
**Michael E. Zieve**- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
- MR Author ID: 614926
- Email: zieve@math.brown.edu
- 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.
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1491853