Factorization of Dickson polynomials over Finite Fields

Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these polynomials in the case that every prime divisor of $n$ divides $q-1$. This result generalizes the results find in Chou, W.S., The Factorization of Dickson polynomials over finite fields. Finite Fields Appl. {\bf3} (1997) 84-96, Fitzgerald R. W., Yucas J. L., Explicit factorization of cyclotomic and Dickson polynomials over finite fields. Arithmetic of Finite Fields. Lecture Notes in Computer Science, vol. {\bf 4547}, pp. 1-10. Springer, Berlin (2007), Tosun, S., Explicit factorizations of generalized Dickson polynomials of order $2^{m}$ via generalized cyclotomic polynomials over finite fields. Finite Fields Appl. {\bf 38} (2016) 40-56 and Tosun, S., Explicit factors of generalized cyclotomic polynomials and generalized Dickson polynomials of order $2^m3$ over finite fields. Discrete Math. 342 (2019) DOI: j.disc.2019.111618