Bivariate factorizations connecting Dickson polynomials and Galois theory

In his Ph.D. Thesis of 1897, Dickson introduced certain permutation polynom ials whose Galois groups are essentially the dihedral groups. These are now called Dickson polynomials of the first kind, to distinguish them from the ir variations introduced by Schur in 1923, which are now called Dickson pol ynomials of the second kind. In the last few decades there have been extens ive investigations of both of these types, which are related to the classic al Chebyshev polynomials. We give new bivariate factorizations involving bo th types of Dickson polynomials. These factorizations demonstrate certain i somorphisms between dihedral groups and orthogonal groups, and lead to the construction of explicit equations with orthogonal groups as Galois groups.