2023/06/19 by Anna-Maurin Graner, Graner, Anna-Maurin · 3 citations
Computer Science · Mathematics · #11T06 #Advanced Mathematical Theories #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2306.11183
openalex publication_date 2023/06/19 · openalex created_date 2023/06/22 · openalex updated_date 2026/07/28
The factorizations of the polynomial Xn-1 and the cyclotomic polynomial Φn over a finite field \mathbb Fq have been studied for a very long time. Explicit factorizations have been given for the case that rad(n)| qw-1 where w=1, w is prime or w is the product of two primes. For arbitrary a∈ \mathbb Fq^∗ the factorization of the polynomial Xn-a is needed for the construction of constacyclic codes. Its factorization has been determined for the case rad(n)| q-1 and for the case that there exist at most three distinct prime factors of n and rad(n)| qw-1 for a prime w. Both polynomials Xn-1 and Xn-a are compositions of the form f(Xn) for a monic irreducible polynomial f∈ \mathbb Fq[X]. The factorization of the composition f(Xn) is known for the case gcd(n, ord(f)⋅ deg(f))=1 and rad(n)| qw-1 for w=1 or w prime. However, there does not exist a closed formula for the explicit factorization of either Xn-1, the cyclotomic polynomial Φn, the binomial Xn-a or the composition f(Xn). 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 Xn-a over \mathbb Fq for any a∈ \mathbb Fq^∗ 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 Xn-1 and of the n-th cyclotomic polynomial Φn 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(Xn) for any irreducible polynomial f∈ \mathbb Fq[X], f≠ X, and any positive integer n such that gcd(n,q)=1 (Theorem 27).