On explicit factors of cyclotomic polynomials over finite fields

We study the explicit factorization of 2nr-th cyclotomic polynomials over finite field Fq where q, r are odd with (r, q) = 1. We show that all irreducible factors of 2nr-th cyclotomic polynomials can be obtained easily from irreducible factors of cyclotomic polynomials of small orders. In particular, we obtain the explicit factorization of 2n5-th cyclotomic polynomials over finite fields and construct several classes of irreducible polynomials of degree 2n−2 with fewer than 5 terms.