By the <Kronecker–Weber theorem>, choose $m$ with
$$
K\subseteq\mathbb Q(\zeta_m).
$$
Restriction gives a quotient
$$
(\mathbb Z/m\mathbb Z)^\times
\twoheadrightarrow G=\operatorname{Gal}(K/\mathbb Q).
$$
Every character $\psi\in\widehat G$ inflates along this quotient and has an associated primitive Dirichlet character $\chi_\psi$, whose conductor may divide $m$. Comparing Euler factors, or applying the factorization of the Artin $L$-function of the regular representation, gives
$$
\boxed{\zeta_K(s)=
\prod_{\psi\in\widehat G}L(s,\chi_\psi).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 3 / c / 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
$$
L\subseteq\mathbb Q(\zeta_m).
$$

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

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Every finite abelian extension of $\mathbb Q$ is contained in a cyclotomic field.


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 Incoming links: Kronecker–Weber theorem