  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.
