= Solution
Multiplying by a positive rational number normalizes the real component and all valuations, leaving the finite unit residue modulo $m$. This identifies
$$
C_{\mathbb Q}/C_{\mathbb Q}(m\infty)
\cong(\mathbb Z/m\mathbb Z)^\times.
$$
Locally,
$$
[\mathbb Z_p^\times:1+p^{n_p}\mathbb Z_p]
=p^{n_p-1}(p-1)
$$
when $n_p>0$. Multiplying gives
$$
\boxed{[C_{\mathbb Q}:C_{\mathbb Q}(m\infty)]
=\prod_{p\mid m}p^{n_p-1}(p-1)
=m\prod_{p\mid m}\left(1-\frac1p\right)
=\varphi(m).}
$$
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