= Solution
The <Kronecker–Weber theorem> says that every finite abelian extension of $\mathbb Q$ lies in some $\mathbb Q(\zeta_m)$.
Indeed, global class field theory assigns to a finite abelian $L/\mathbb Q$ the open <norm group of an abelian extension> $N_{L/\mathbb Q}C_L$. It contains a congruence subgroup $C_{\mathbb Q}(m\infty)$. By part (e), this is the norm group of $\mathbb Q(\zeta_m)$. The inclusion-reversing class-field correspondence therefore gives
$$
L\subseteq\mathbb Q(\zeta_m).
$$
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