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 mth 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 mth cyclotomic polynomial, we have equality and in fact f is the mth cyclotomic polynomial, which must therefore be irreducible.

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