Dickson polynomials of both kinds over finite fields are explicitly factored into irreducible factors when rad(n) divides q-1, and rad(n+1) divides q-1 for the second kind.
Finite fields and Their Applications 24 95-104 (2013)
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Factorization of Dickson polynomials over Finite Fields
Dickson polynomials of both kinds over finite fields are explicitly factored into irreducible factors when rad(n) divides q-1, and rad(n+1) divides q-1 for the second kind.