The kth Dickson polynomial of the first kind, D k (x)∈ Z [x], is determined by the formula: D k (u+ 1/u)= u k+ 1/u k, 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 D k (x) is a permutation polynomial on F p n, p prime, if and only if GCD (k, p 2 n− 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 F p n 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∈ F q×: a and 4− a are nonsquares}= 2.