Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields

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}.