Closed formulas for the factorization of $X^n-1$, the $n$-th cyclotomic polynomial, $X^n-a$ and $f(X^n)$ over a finite field for arbitrary positive integers $n$

Abstract

The factorizations of the polynomial $X^n-1$ and the cyclotomic polynomial ${\Phi }_n$ over a finite field ${\mathbb{F}}_q$ have been studied for a very long time. Explicit factorizations have been given for the case that $\mathrm{rad}(n)\mid q^w-1$ where $w=1$, $w$ is prime or $w$ is the product of two primes. For arbitrary $a\in {\mathbb{F}}_q^{\ast }$ the factorization of the polynomial $X^n-a$ is needed for the construction of constacyclic codes. Its factorization has been determined for the case $\mathrm{rad}(n)\mid q-1$ and for the case that there exist at most three distinct prime factors of $n$ and $\mathrm{rad}(n)\mid q^w-1$ for a prime $w$. Both polynomials $X^n-1$ and $X^n-a$ are compositions of the form $f(X^n)$ for a monic irreducible polynomial $f\in {\mathbb{F}}_q[X]$. The factorization of the composition $f(X^n)$ is known for the case $\gcd (n,\mathrm{ord}(f)\cdot \mathrm{deg}(f))=1$ and $\mathrm{rad}(n)\mid q^w-1$ for $w=1$ or $w$ prime. However, there does not exist a closed formula for the explicit factorization of either $X^n-1$, the cyclotomic polynomial ${\Phi }_n$, the binomial $X^n-a$ or the composition $f(X^n)$. Without loss of generality we can assume that $\gcd (n,q)=1$. Our main theorem, Theorem 18, is a closed formula for the factorization of $X^n-a$ over ${\mathbb{F}}_q$ for any $a\in {\mathbb{F}}_q^{\ast }$ and any positive integer $n$ such that $\gcd (n,q)=1$. From our main theorem we derive one closed formula each for the factorization of $X^n-1$ and of the $n$-th cyclotomic polynomial ${\Phi }_n$ for any positive integer $n$ such that $\gcd (n,q)=1$ (Theorem 2.5 and Theorem 2.6). Furthermore, our main theorem yields a closed formula for the factorization of the composition $f(X^n)$ for any irreducible polynomial $f\in {\mathbb{F}}_q[X]$, $f\ne X$, and any positive integer $n$ such that $\gcd (n,q)=1$ (Theorem 27).

We added factorizations of $X^n-1$ and the $n$-th cyclotomic polynomial. We improved the selection of the parameters for our main theorem, gave a more thorough proof for it and corrected the choice of the representative system for the case $gcd(s_1,s_2)>1$. We included a reference to [WY18]. In Proposition 6 we corrected the choice of $r$ for the case $a=1$