Exercise 9.2.2

Proof. Write ${\tilde{H}}_f$ the subgroup corresponding to $H_f$ by the isomorphism $\mathrm{Gal}(\mathbb{Q}({\zeta }_p)∕\mathbb{Q})\simeq {(\mathbb{Z}∕\mathit{p\mathbb{Z}})}^{\text{∗}}$. Then

and

is the fixed field of ${\tilde{H}}_f$, with $\mathbb{Q}\subset L_f\subset \mathbb{Q}({\zeta }_p)$.

(a)

As $G=\mathrm{Gal}(\mathbb{Q}({\zeta }_p)∕\mathbb{Q})$ is Abelian ( $G$ is cyclic since ${(\mathbb{Z}∕\mathit{p\mathbb{Z}})}^{\text{∗}}\simeq G$ is cyclic for prime $p$), so every subgroup of $G$ is normal, therefore $\mathbb{Q}\subset L_f$ is a Galois extension (Theorem 7.3.2).

Moreover, by the Galois correspondence (Theorem 7.3.1), $[L_f:\mathbb{Q}]=(G:{\tilde{H}}_f),$ and $(G:{\tilde{H}}_f)=({(\mathbb{Z}∕\mathit{p\mathbb{Z}})}^{\text{∗}}:H_f)=(p-1)∕f=e$, so

$$[L_f:\mathbb{Q}]=e.$$

$L_f$ is a Galois extension of $\mathbb{Q}$ of degree $e$.

(b)

By Exercise 1, $f\mid f^{\prime }⟺H_f\subset H_{f^{\prime }}$. As the Galois correspondence is order reversing, $$f\mid f^{\prime }⟺H_f\subset H_{f^{\prime }}⟺{\tilde{H}}_f\subset {\tilde{H}}_{f^{\prime }}⟺L_f\supset L_{f^{\prime }}.$$

(c)

Let $f,f^{\prime }$ be positive divisors of $p-1$ such that $f\mid f^{\prime }$. Since $G$ is Abelian, $L_{f^{\prime }}\subset L_f$ is a Galois extension, and by Theorem 7.3.2, $$\mathrm{Gal}(L_f∕L_{f^{\prime }})\simeq \mathrm{Gal}(\mathbb{Q}({\zeta }_p)∕L_{f^{\prime }})∕\mathrm{Gal}(\mathbb{Q}({\zeta }_p)∕L_f)={\tilde{H}}_{f^{\prime }}∕{\tilde{H}}_f\simeq H_{f^{\prime }}∕H_f.$$

As $H_{f^{\prime }}$ is cyclic of order $f^{\prime }$, the quotient group $H_{f^{\prime }}∕H_f$ is itself cyclic, of order $f^{\prime }∕f$.

Conclusion:

$\mathrm{Gal}(L_f∕L_{f^{\prime }})$ is cyclic of order $f^{\prime }∕f$.