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
$$
L\subseteq\mathbb Q(\zeta_m).
$$

Solution

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

{title2=$N_{L/K}C_L$}

For a finite abelian extension $L/K$, the norm group is the image of the idele-class norm $C_L\to C_K$. Class field theory gives $C_K/N_{L/K}C_L\cong\operatorname{Gal}(L/K)$.


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 Incoming links: Norm group of an abelian extension