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), 28712887
MSC (1991):
Primary 12E05, 12F10, 14H30, 20D06, 20G40, 20E22
Published electronically:
February 28, 2000
MathSciNet review:
1491853
Fulltext PDF Free Access
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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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 S. S. Abhyankar, Nice equations for nice groups, Israel Journal of Mathematics 88 (1994), 124. MR 96f:12003
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 S. S. Abhyankar, Fundamental group of the affine line in positive characteristic, Proceedings of the 1992 International Colloquium on Geometry and Analysis, Tata Institute of Fundamental Research, Bombay (1995), 126. MR 97b:14034
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 S. S. Abhyankar, Again nice equations for nice groups, Proceedings of the American Mathematical Society 124 (1996), 29672976. MR 96m:12004
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 S. S. Abhyankar, Factorizations over finite fields, Finite Fields and Applications, London Mathematical Society, Lecture Note Series 233 (1996), 121. MR 98c:11130
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 S. D. Cohen and R. Matthews, Monodromy groups of classical families over finite fields, Finite Fields and Applications, London Mathematical Society, Lecture Note Series 233 (1996), 5968. MR 98f:11131
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 L. E. Dickson, The analytic presentation of substitutions on a power of a prime number of letters with a discussion of the linear group, Annals of Mathematics 11 (1897), 65120.
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 L. E. Dickson, Linear Groups, Teubner, 1901.
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 M. Liebeck, Characterization of classical groups by orbit sizes on the natural module, Proceedings of the American Mathematical Society 124 (1996), 29612966. MR 97e:20068
 [LMT]
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 [Sch]
 I. Schur, Über den Zusammenhang zwischen einemem Problem der Zahlentheorie und einem Satz über algebraische Funktionen, Sitzungsber. Akad. Wiss. Berlin (1923), 123134.
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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/S0002994700022716
PII:
S 00029947(00)022716
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 9101424 and NSA grant MDA 9049710010. 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
