Explicit factorizations of cyclotomic polynomials over finite fields

Let q be a prime power and let $${\mathbb {F}}_q$$Fq be a finite field with q elements. This paper discusses the explicit factorizations of cyclotomic polynomials over $$\mathbb {F}_q$$Fq. Previously, it has been shown that to obtain the factorizations of the $$2^{n}r$$2nrth cyclotomic polynomials, one only need to solve the factorizations of a finite number of cyclotomic polynomials. This paper shows that with an additional condition that $$q\equiv 1 \pmod p$$qź1(modp), the result can be generalized to the $$p^{n}r$$pnrth cyclotomic polynomials, where p is an arbitrary odd prime. Applying this result we discuss the factorization of cyclotomic polynomials over finite fields. As examples we give the explicit factorizations of the $$3^{n}$$3nth, $$3^{n}5$$3n5th and $$3^{n}7$$3n7th cyclotomic polynomials.