Analytic proof that cyclotomic polynomials are irreducible

Apparently this is due to Kronecker – the idea is to prove that cyclotomic polynomials are irreducible indirectly using the following two analytic facts:

Fact 1 (Dirichlet’s theorem, special case): The Dirichlet density of primes which are congruent to $1$ modulo $m$ is precisely $\frac{1}{\phi(m)}$ .

Fact 2 (due to Kronecker, but a super special case of Cebotarev’s density theorem): The set of primes which split completely in an extension $L/\mathbb{Q}$ has Dirichlet density $\frac{1}{[L:\mathbb{Q}]}$ .

And now we just stick these facts together. Let $L=\mathbb{Q}(\zeta_m)$ and take $f$ to be the minimal polynomial of $\zeta_m$ . Then a rational prime $p$ splits iff $f$ splits modulo $p$ , which only happens (assuming $p$ does not divide $m$ ) if there is a primitive $m$ th root of unity in $\mathbb{F}_p$ . For this to be the case we must have $m|p-1$ . Thus by fact 1, at most density $\frac{1}{\phi(m)}$ primes split in this extension. Hence by fact 2, $deg f = [L:\mathbb{Q}] \geq \phi(m)$ , and since the RHS is the degree of the $m$ th cyclotomic polynomial, we have equality and in fact $f$ is the $m$ th cyclotomic polynomial, which must therefore be irreducible.