Proposition 6

— Let $p$ be the characteristic exponent of $K$ and let $n \geq 1$ be an integer prime to $p$. For the polynomial $\Phi_n(X)$ to be irreducible in $K[X]$ it is necessary and sufficient that the homomorphism $\chi_n : \mathrm{Gal}(K_s/K) \to (\mathbf{Z}/n\mathbf{Z})^*$ should be surjective.

By Lemma 3 we have $R_n(K) = K(\zeta)$ for each root $\zeta$ of $\Phi_n(X)$ and hence $\Phi_n(X)$ is irreducible in $K[X]$ if and only if the degree $\varphi(n)$ of $\Phi_n(X)$ is equal to $[R_n(K):K]$. Further, the Galois group of $R_n(K)$ over $K$ is of order $[R_n(K):K]$ and it is isomorphic to the subgroup of $(\mathbf{Z}/n\mathbf{Z})^*$ which is the image of $\chi_n$. Now Prop. 6 follows from the fact that $(\mathbf{Z}/n\mathbf{Z})^*$ is of order $\varphi(n)$.