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

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 ${\Phi }_p(t)=1+t+\cdots +t^{p-1}$ 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.