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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 $n$

Anna-Maurin Graner

Abstract

The factorizations of the polynomial $X^n-1$ and the cyclotomic polynomial $Φ_n$ 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 $Φ_n$, 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 $Φ_n$ 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).

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 and the cyclotomic polynomial over a finite field have been studied for a very long time. Explicit factorizations have been given for the case that where , is prime or is the product of two primes. For arbitrary the factorization of the polynomial is needed for the construction of constacyclic codes. Its factorization has been determined for the case and for the case that there exist at most three distinct prime factors of and for a prime . Both polynomials and are compositions of the form for a monic irreducible polynomial . The factorization of the composition is known for the case and for or prime. However, there does not exist a closed formula for the explicit factorization of either , the cyclotomic polynomial , the binomial or the composition . Without loss of generality we can assume that . Our main theorem, Theorem 18, is a closed formula for the factorization of over for any and any positive integer such that . From our main theorem we derive one closed formula each for the factorization of and of the -th cyclotomic polynomial for any positive integer such that (Theorem 2.5 and Theorem 2.6). Furthermore, our main theorem yields a closed formula for the factorization of the composition for any irreducible polynomial , , and any positive integer such that (Theorem 27).
Paper Structure (7 sections, 25 theorems, 53 equations)

This paper contains 7 sections, 25 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Structure of this paper:
  3. The factorization of $X^n-a$
  4. The factorization of $X^n-a$ for positive integers $n$ such that $\mathrm{rad}(n)\mid q-1$ and ($4\nmid n$ or $q\equiv 1 \mod 4$)
  5. The factorization of $X^n-a$ for positive integers $n$ such that $\gcd(n,q)=1$
  6. The factorization of $X^n-1$ and of the cyclotomic polynomial $\Phi_n$ for positive integers $n$ such that $\gcd(n,q)=1$
  7. The factorization of $f(X^n)$

Key Result

Theorem 1

Let $f\in \mathbb F_q[X]$ be an irreducible polynomial of degree $k$ and $\alpha\in \mathbb F_{q^{k}}$ be a root of $f$. Further, let $Q = \frac{g}{h} \in \mathbb F_q(X)$ be a rational function over $\mathbb F_q$. Then the polynomial $f^Q$ is irreducible over $\mathbb F_q$ if and only if the polynom

Theorems & Definitions (50)

  • Theorem 1: Cohen1969
  • Theorem 2: Kyuregyan2011
  • Theorem 3: Mullin2010
  • proof : Proof of \ref{['Mullin2010: Lemma 13']}
  • Lemma 4
  • proof
  • Proposition 5
  • Theorem 1.1: Butler1955
  • Remark 2.1
  • Proposition 6
  • ...and 40 more