Explicit factorizations, into a product of irreducible polynomials, over Fq of the
cyclotomic polynomials Q2n(x) are given in [4] when q ≡ 1 (mod 4). The case
q ≡ 3 (mod 4) is done in [5]. Here we give factorizations of Q2nr(x) where r
is prime and q ≡ ±1 (mod r). In particular, this covers Q2n3(x) for all Fq of
characteristic not 2, 3. We apply this to get explicit factorizations of the first
and second kind Dickson polynomials of order 2n3 and 2n3 − 1 respectively.
Explicit factorizations of certain Dickson polynomials have been used to compute
Brewer sums [1]. But our basic motivation is curiosity, to see what factors
arise. Of interest then is how the generalized Dickson polynomials Dn(x, b) arise
in the factors of the cyclotomic polynomials and how the Dickson polynomials
of the first kind appear in the factors of both kinds of Dickson polynomials.