 Solution

The primitive root ${\zeta }_n$ is a zero of the cyclotomic polynomial ${\Phi }_n$, of degree $\phi (n)$. Its minimal polynomial over $K$ divides ${\Phi }_n$, sowhere divisibility also follows from the embedding of the Galois group of the cyclotomic field into $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$. Over $\mathbb{Q}$, Gauss's irreducibility theorem for cyclotomic polynomials says that ${\Phi }_n$ is irreducible. Therefore