Irreducibility of the cyclotomic polynomials over the rationals

Statement

For every integer $n \ge 1$, the cyclotomic polynomial $\Phi_n$ is irreducible in $\mathbb{Q}[x]$.

Remarks

For $n = p$ prime, Eisenstein's criterion applied to $\Phi_p(x + 1)$ with the prime $p$ proves the statement. The general case is harder. One proof shows that if $\zeta$ is a root of an irreducible factor of $\Phi_n$, then so is $\zeta^p$ for each prime $p \nmid n$. A consequence: $\Phi_n$ is the minimal polynomial over $\mathbb{Q}$ of each primitive $n$-th root of unity. So the degree of $\mathbb{Q}(e^{2\pi i/n})$ over $\mathbb{Q}$ is $\varphi(n)$.

Explanations

L1 human

The polynomial $\Phi_n$ cannot be split into smaller polynomials with rational coefficients. So all the primitive $n$-th roots of unity are conjugate over the rationals: no rational polynomial equation separates them.