Abstract
The kth Dickson polynomial of the first kind, Dk(x)∈Z[x], is determined by the formula: Dk(u+1/u)=uk+1/uk, where k≥0 and u is an indeterminate. These polynomials are closely related to Chebyshev polynomials and have been widely studied. Leonard Eugene Dickson proved in 1896 that Dk(x) is a permutation polynomial on Fpn, p prime, if and only if GCD(k,p2n−1)=1, and his result easily carries over to Chebyshev polynomials when p is odd. This article continues on this theme, as we find special subsets of Fpn that are stabilized or permuted by Dickson or Chebyshev polynomials. Our analysis also leads to a factorization formula for Dickson and Chebyshev polynomials and some new results in elementary number theory. For example, we show that if q is an odd prime power, then ∏{a∈Fq×:a and 4−a are nonsquares}=2.
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