Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields

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 ${\mathrm{\Phi }}_{2^{n}r}$ over ${\mathbb{F}}_q$ where both $r\ge 3$ 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 ${\mathrm{\Phi }}_r$ over ${\mathbb{F}}_q$ is given to us as a known. Let $n = p_1^{e_1}p_2^{e_2}… p_s^{e_s}$bethefactorizationof$n \in $\mathbb{N}$$ into powers of distinct primes $p_i,1\le i\le s$. 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 ${\mathrm{\Phi }}_n$ from the factors of each ${\mathrm{\Phi }}_{p_i^{e_i}}$. We also demonstrate how to obtain the factorization of ${\mathrm{\Phi }}_{mn}$ from the factorization of ${\mathrm{\Phi }}_n$ when $q$ is a primitive root modulo $m$ and $\gcd (m,n)=\gcd (\varphi (m),{\text{ord}}_n(q))=1$. Here $\varphi$ is the Euler’s totient function, and ${\text{ord}}_n(q)$ 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}.