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

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 $f\in \mathbb F_q[X]$, $f\neq X$, and any positive integer $n$ such that $\gcd(n,q)=1$ (Theorem 27).Comment: 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

Constructing irreducible polynomials recursively with a reverse composition method

We suggest a construction of the minimal polynomial $m_{\beta^k}$ of $\beta^k\in \mathbb F_{q^n}$ over $\mathbb F_q$ from the minimal polynomial $f= m_\beta$ for all positive integers $k$ whose prime factors divide $q-1$. The computations of our construction are carried out in $\mathbb F_q$. The key observation leading to our construction is that for $k \mid q-1$ holds $$m_{\beta^k}(X^k) = \prod_{j=1}^{\frac kt} \zeta_k^{-jn} f (\zeta_k^j X),$$ where $t= \max \{m\mid \gcd(n,k): f (X) = g (X^m), g \in \mathbb F_q[X]\}$ and $\zeta_{k}$ is a primitive $k$-th root of unity in $\mathbb F_q$. The construction allows to construct a large number of irreducible polynomials over $\mathbb F_q$ of the same degree. Since different applications require different properties, this large number allows the selection of the candidates with the desired properties

On the bijectivity of the map χ\chi

We prove that for $n>1$ the map $\chi:\mathbb{F}_q^n \to \mathbb{F}_q^n$, defined by $y=\chi(x)$ with $y_i = x_i + x_{i+2}\cdot(1+x_{i+1})$ for $1\leq i \leq n$, is bijective if and only if $q=2$ and $n$ is odd, as it was conjectured by Schoone and Daemen in 2023