Corollary

— The degree of $R_n$ over $K$ is the least integer $f \geq 1$ such that $q^f \equiv 1 \pmod{n}$. For the cyclotomic polynomial $\Phi_n$ to be irreducible over $K$ it is necessary and sufficient that the group $(\mathbf{Z}/n\mathbf{Z})^*$ should be generated by the residue class of $q$ modulo $n$.