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. The reciprocals of these irreducible polynomials are irreducible polynomials of the form $x^{2^{n-2}}+g(x)$ such that the degree of $g(x)$ is small ($\le 4$), which could have potential applications as mentioned by Gao, Howell, and Panario in \cite{GaoHowellPanario}.