On prime values of cyclotomic polynomials

TL;DR

This paper investigates conditions under which cyclotomic polynomials evaluated at powers of x remain irreducible, providing new insights into their algebraic properties and factorization behavior.

Contribution

It introduces several approaches to determine necessary and sufficient conditions for the irreducibility of _k(x^n), advancing understanding of cyclotomic polynomial evaluations.

Findings

Identified key conditions for irreducibility of _k(x^n)

Developed multiple methods for analyzing polynomial irreducibility

Enhanced theoretical understanding of cyclotomic polynomial evaluations

Abstract

We present several approaches on finding necessary and sufficient conditions on $n$ so that $\Phi_k(x^n)$ is irreducible where $\Phi_k$ is the $k$-th cyclotomic polynomial.