Exercise 7.3.14

Proof. Suppose that $L\subset \mathbb{Q}({\zeta }_n)$, where ${\zeta }_n=e^{2\mathit{\pi i}∕n}$. The Exercise 6.2.4 prove the existence of an injective group homomorphism, given by

Consequently $G=\mathrm{Gal}(\mathbb{Q}({\zeta }_n)∕\mathbb{Q})$ is isomorphic to a subgroup of ${(\mathbb{Z}∕\mathit{n\mathbb{Z}})}^{\text{∗}}$, so $G$ is Abelian. As all subgroups of an Abelian group are normal, $H=\mathrm{Gal}(\mathbb{Q}({\zeta }_n)∕L)$ is a normal subgroup of $G$, therefore (Theorem 7.2.5) $\mathbb{Q}\subset L$ is a Galois extension, a fortiori a normal extension, and $\mathrm{Gal}(L∕\mathbb{Q})\simeq \mathrm{Gal}(\mathbb{Q}({\zeta }_n)∕\mathbb{Q})∕\mathrm{Gal}(\mathbb{Q}({\zeta }_n)∕L)$ is isomorphic to a quotient group of an Abelian group, so is Abelian: the implication (b) $\Rightarrow$(a) of Theorem 6.5.5 is proved. □