An Infinite, Two-parameter Family of Polynomials with Factorization Similar to X^m-1

2021/11/29 by Vanchinathan, P, M, Krithika
#11S20 #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2111.14389

Abstract

For a suitable irreducible base polynomial f(x)∈ Z[x] of degree k, a family of polynomials F_m(x) depending on f(x) is constructed with the properties: (i) there is exactly one irreducible factor Φ_d,f(x) for F_m(x) for each divisor d of m; (ii) deg (Φ_d,f(x))=φ(d)\cdotdeg (f) generalizing the factorization of x^m-1 into cyclotomic polynomials; (iii) when the base polynomial f(x) = x-1 this F_m(x) coincides with x^m-1. As an application, irreducible polynomials of degree 12, 24, 24 are constructed having Galois groups of order matching their degrees and isomorphic to S₃ ⊕ C₂ , S₃ ⊕ C₂⊕ C₂ and S₃ ⊕ C₄ respectively.

Related

• An irreducible class of polynomials over integers
• Factoring polynomials of the form f(xⁿ)∈ \mathbbF_q[x]
• Constructing irreducible polynomials recursively with a reverse composition method
• Factorization of a class of composed polynomials
• Closed formulas for the factorization of Xⁿ-1, the n-th cyclotomic polynomial, Xⁿ-a and f(Xⁿ) over a finite field for arbitrary positive integers n