The assumed inclusion and the class-field norm-index theorem give
$$
C_{\mathbb Q}(m\infty)
\subseteq N_{\mathbb Q(\zeta_m)/\mathbb Q}C_{\mathbb Q(\zeta_m)}
\subseteq C_{\mathbb Q}.
$$
The outer subgroup has index $\varphi(m)$ by part (d), while
$$
[C_{\mathbb Q}:N_{\mathbb Q(\zeta_m)/\mathbb Q}C_{\mathbb Q(\zeta_m)}]
=[\mathbb Q(\zeta_m):\mathbb Q]
=\varphi(m).
$$
Two nested subgroups of the same finite index are equal. Thus
$$
\boxed{C_{\mathbb Q}(m\infty)
=N_{\mathbb Q(\zeta_m)/\mathbb Q}C_{\mathbb Q(\zeta_m)}.}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 4 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-123/4/e/solution