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 modulo is precisely .

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 has Dirichlet density .

And now we just stick these facts together. Let and take to be the minimal polynomial of . Then a rational prime splits iff splits modulo , which only happens (assuming does not divide ) if there is a primitive th root of unity in . For this to be the case we must have . Thus by fact 1, at most density primes split in this extension. Hence by fact 2, , and since the RHS is the degree of the th cyclotomic polynomial, we have equality and in fact *is* the th cyclotomic polynomial, which must therefore be irreducible.

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