An alternative proof for the irreducibility of the p-th cyclotomic polynomial

TL;DR

This paper presents an alternative proof for the irreducibility of the p-th cyclotomic polynomial, using determinants and a lemma by Kronecker, offering a different perspective from the traditional Eisenstein criterion.

Contribution

It introduces a novel proof method for the irreducibility of cyclotomic polynomials based on determinants and Kronecker's lemma, expanding the theoretical understanding.

Findings

Provides an alternative proof for the irreducibility of  p-th cyclotomic polynomial

Utilizes determinants and Kronecker's lemma in the proof

Offers a new perspective beyond Eisenstein's criterion

Abstract

Let $p$ be a prime number. As a standard application of the irreducibility criterion of Eisenstein, it is well known that the $p$-th cyclotomic polynomial $\Phi_p(t)=1+t+\dots+t^{p-1}$ is the minimal polynomial of $e^{2\pi i/p}$ over $\mathbb{Q}$. This note provides an alternative proof, utilizing determinants to prove a lemma due to Kronecker.