On explicit factors of cyclotomic polynomials over finite fields

Abstract

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