Lemma 3

The roots of $\Phi_n$ in $K_s$ are the primitive $n$-th roots of unity.

Denote by $S$, the set of roots of $\Phi_n$ in $K_s$. By Formula (6), the set $\mu_n(K_s)$ is the union of the $S_d$ for $d$ dividing $n$. Every primitive $n$-th root of unity thus belongs to $S_n$ and the lemma now follows from Prop. 4 (V, p. 81).