Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields

Details Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields

2011-09-22

arxiv.org/abs/1109.4693v1

Mathematics

Number Theory

24 pages

Scientific paper

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}.

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