On explicit factors of Cyclotomic polynomials over finite fields

TL;DR

This paper provides explicit factorizations of certain cyclotomic polynomials over finite fields, enabling easier construction of irreducible polynomials with applications in finite field theory and related areas.

Contribution

It introduces a method to derive all irreducible factors of $2^n r$-th cyclotomic polynomials from small order factors, including explicit factorizations for $2^n 5$-th cases.

Findings

Explicit factorization of $2^n 5$-th cyclotomic polynomials over finite fields.

Construction of irreducible polynomials of degree $2^{n-2}$ with fewer than 5 terms.

Reciprocal polynomials with small degree $g(x)$, useful for applications.

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 ($\leq 4$), which could have potential applications as mentioned by Gao, Howell, and Panario in \cite{GaoHowellPanario}.