Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields

TL;DR

This paper provides explicit factorizations of cyclotomic polynomials over finite fields for new cases, extending previous results, and introduces methods to derive factors of composite indices from prime power factors.

Contribution

It offers explicit factorization formulas for cyclotomic polynomials over finite fields for broader cases and introduces new constructions of irreducible polynomials.

Findings

Explicit factorization of $\

Methods to derive factors of $\

Abstract

Let $q = p^s$ be a power of a prime number $p$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. In this paper we obtain the explicit factorization of the cyclotomic polynomial $\Phi_{2^nr}$ over $\mathbb{F}_q$ where both $r \geq 3$ and $q$ are odd, $\gcd(q,r) = 1$, and $n\in \mathbb{N}$. Previously, only the special cases when $r = 1,\ 3,\ 5$ had been achieved. For this we make the assumption that the explicit factorization of $\Phi_r$ over $\mathbb{F}_q$ is given to us as a known. Let $n = p_1^{e_1}p_2^{e_2}... p_s^{e_s}$ be the factorization of $n \in \mathbb{N}$ into powers of distinct primes $p_i,\ 1\leq i \leq s$. In the case that the orders of $q$ modulo all these prime powers $p_i^{e_i}$ are pairwise coprime we show how to obtain the explicit factors of $\Phi_{n}$ from the factors of each $\Phi_{p_i^{e_i}}$. We also demonstrate how to obtain the factorization of $\Phi_{mn}$ from the factorization of $\Phi_n$ when $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),\ord_n(q)) = 1$. Here $\phi$ is the Euler's totient function, and $\ord_n(q)$ denotes the multiplicative order of $q$ modulo $n$. Moreover, we present the construction of a new class of irreducible polynomials over $\mathbb{F}_q$ and generalize a result due to Varshamov (1984) \cite{Varshamov}.