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

TL;DR

This paper introduces a new family of polynomials generalizing cyclotomic polynomials, with explicit factorization properties linked to a base polynomial, and constructs specific irreducible polynomials with prescribed Galois groups.

Contribution

It constructs an infinite two-parameter family of polynomials with explicit factorization properties generalizing $x^m-1$, extending cyclotomic polynomial theory.

Findings

Family generalizes cyclotomic polynomial factorization

Explicit construction of irreducible polynomials with specific Galois groups

Provides new tools for polynomial factorization and Galois theory

Abstract

For a suitable irreducible \textit{base} polynomial $f(x)\in \mathbf{Z}[x]$ 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 $\Phi_{d,f}(x)$ for $F_m(x)$ for each divisor $d$ of $m$; (ii) deg $(\Phi_{d,f}(x))=\varphi(d)\cdot\mathrm{deg} (f)$ 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 $S_3 \oplus C_2 , S_3 \oplus C_2\oplus C_2$ and $S_3 \oplus C_4$ respectively.