[proofplan] We prove that every finite abelian extension $K/\mathbb{Q}$ is contained in a cyclotomic field. The argument has three stages. First, we identify the finite set $S$ of primes ramifying in $K$ and recall the structure of cyclotomic extensions. Second, at each prime $p \in S$ we pass to the completion $K_\mathfrak{P}/\mathbb{Q}_p$ and apply the Local Kronecker-Weber Theorem to embed it into a local cyclotomic extension. Third, we handle unramified primes by exploiting the structure of unramified extensions of $\mathbb{Q}_p$ and assemble all the local containments into the global containment $K \subseteq \mathbb{Q}(\zeta_n)$ via the Artin reciprocity law. [/proofplan]

[step:Reduce to finite extensions and identify the ramified primes]Let $K/\mathbb{Q}$ be a finite abelian extension (every abelian extension of $\mathbb{Q}$ is a directed union of finite abelian sub-extensions, so it suffices to treat the finite case). By the [Fundamental Theorem of Galois Theory](/theorems/1274), $K/\mathbb{Q}$ is Galois with $G = \operatorname{Gal}(K/\mathbb{Q})$ a finite abelian group. Let $S = \{p_1, \ldots, p_r\}$ denote the set of rational primes that ramify in $K/\mathbb{Q}$. By the [Ramification and Discriminant](/theorems/1601) theorem, a prime $p$ ramifies in $K$ if and only if $p \mid \Delta_K$, where $\Delta_K \in \mathbb{Z} \setminus \{0\}$ is the discriminant of $K$. Since $\Delta_K$ is a non-zero integer, $S$ is finite.[/step]

[guided]Every abelian extension of $\mathbb{Q}$ is a directed union of finite abelian sub-extensions. If each finite abelian sub-extension $K$ is contained in some $\mathbb{Q}(\zeta_{n_K})$, then the full extension is contained in $\bigcup_K \mathbb{Q}(\zeta_{n_K})$, which is itself a union of cyclotomic fields. So it suffices to prove the theorem for finite abelian extensions. Let $K/\mathbb{Q}$ be a finite abelian extension. Since $K/\mathbb{Q}$ is Galois (abelian extensions are Galois by definition) with abelian Galois group $G = \operatorname{Gal}(K/\mathbb{Q})$, we can apply the full machinery of algebraic number theory. We need to identify which rational primes ramify in $K$. A prime $p$ ramifies in $K$ if there exists a prime ideal $\mathfrak{P}$ of $\mathcal{O}_K$ lying above $p$ with ramification index $e_{\mathfrak{P}/p} > 1$, where the ramification index $e_{\mathfrak{P}/p}$ is the exponent of $\mathfrak{P}$ in the factorization of $p\mathcal{O}_K$. The [Ramification and Discriminant](/theorems/1601) theorem states that $p$ ramifies in $K$ if and only if $p \mid \Delta_K$. Since $\Delta_K \in \mathbb{Z} \setminus \{0\}$, only finitely many primes ramify. Denote this finite set by $S = \{p_1, \ldots, p_r\}$. Why does the finiteness of $S$ matter? The strategy is to handle each ramified prime separately using local methods. With finitely many primes, we can take a finite product (or lcm) of the local moduli to construct a single cyclotomic field containing $K$.[/guided]

[step:Recall the structure and ramification of cyclotomic fields]By the [Galois Group of Cyclotomic Extensions Over Q](/theorems/1280) theorem, for every $m \ge 1$, the extension $\mathbb{Q}(\zeta_m)/\mathbb{Q}$ is Galois with \begin{align*} \operatorname{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q}) \cong (\mathbb{Z}/m\mathbb{Z})^\times, \end{align*} which is abelian of order $\varphi(m)$. The primes that ramify in $\mathbb{Q}(\zeta_m)$ are exactly the primes dividing $m$. For a prime power $p^k$ with $k \ge 1$, the extension $\mathbb{Q}(\zeta_{p^k})/\mathbb{Q}$ is totally ramified at $p$: the unique prime of $\mathbb{Z}[\zeta_{p^k}]$ above $p$ is the principal ideal $(1 - \zeta_{p^k})$, with ramification index $\varphi(p^k) = p^{k-1}(p-1) = [\mathbb{Q}(\zeta_{p^k}):\mathbb{Q}]$. At all primes $\ell \neq p$, the extension $\mathbb{Q}(\zeta_{p^k})/\mathbb{Q}$ is unramified. By the Chinese Remainder Theorem, if $m = \prod_{i} p_i^{k_i}$ is the prime factorization of $m$, then \begin{align*} (\mathbb{Z}/m\mathbb{Z})^\times \cong \prod_{i} (\mathbb{Z}/p_i^{k_i}\mathbb{Z})^\times, \end{align*} and $\mathbb{Q}(\zeta_m)$ is the compositum of the fields $\mathbb{Q}(\zeta_{p_i^{k_i}})$.[/step]

[guided]The cyclotomic fields are the building blocks of the proof. The [Galois Group of Cyclotomic Extensions Over Q](/theorems/1280) theorem gives us an explicit description: $\operatorname{Gal}(\mathbb{Q}(\zeta_m)/\mathbb{Q}) \cong (\mathbb{Z}/m\mathbb{Z})^\times$ via the map sending $\sigma_a$ (defined by $\sigma_a(\zeta_m) = \zeta_m^a$) to $a \bmod m$. This group is abelian of order $\varphi(m)$, so every cyclotomic extension is abelian over $\mathbb{Q}$. The theorem we are proving is the converse: every abelian extension arises this way. The ramification of cyclotomic fields is completely understood. For a prime power $p^k$ with $k \ge 1$, the minimal polynomial of $\zeta_{p^k}$ over $\mathbb{Q}$ is the $p^k$-th cyclotomic polynomial $\Phi_{p^k}(x) = x^{p^{k-1}(p-1)} + x^{p^{k-1}(p-2)} + \cdots + x^{p^{k-1}} + 1$, and the discriminant calculation shows that $p$ is the only prime that ramifies in $\mathbb{Q}(\zeta_{p^k})$. The unique prime above $p$ is $(1 - \zeta_{p^k})$, with ramification index $\varphi(p^k) = p^{k-1}(p-1) = [\mathbb{Q}(\zeta_{p^k}):\mathbb{Q}]$, so the extension is totally ramified at $p$ and unramified everywhere else. By the Chinese Remainder Theorem, for $m = \prod_{i} p_i^{k_i}$, we have $(\mathbb{Z}/m\mathbb{Z})^\times \cong \prod_{i} (\mathbb{Z}/p_i^{k_i}\mathbb{Z})^\times$. Correspondingly, $\mathbb{Q}(\zeta_m) = \prod_i \mathbb{Q}(\zeta_{p_i^{k_i}})$ (compositum), and the ramification at each prime $p_i$ is controlled solely by the factor $\mathbb{Q}(\zeta_{p_i^{k_i}})$. This complete understanding of cyclotomic ramification will allow us to match the local extensions of $K$ with local cyclotomic extensions prime by prime.[/guided]

[step:Define completions and decomposition groups, then apply the Local Kronecker-Weber Theorem at each ramified place]We introduce the local notation. For a prime $p$ and a prime ideal $\mathfrak{P}$ of $\mathcal{O}_K$ lying above $p$, define: - The **completion** $K_\mathfrak{P}$: the completion of $K$ with respect to the $\mathfrak{P}$-adic absolute value $|\cdot|_\mathfrak{P}$. This is a finite extension of $\mathbb{Q}_p$ of degree $[K_\mathfrak{P}:\mathbb{Q}_p] = e_{\mathfrak{P}/p} \cdot f_{\mathfrak{P}/p}$, where $e_{\mathfrak{P}/p}$ is the ramification index and $f_{\mathfrak{P}/p}$ is the residue degree. - The **decomposition group** $D_{\mathfrak{P}/p} := \{\sigma \in G : \sigma(\mathfrak{P}) = \mathfrak{P}\} \le G$: the stabilizer of $\mathfrak{P}$ under the action of $G$ on primes above $p$. By the Decomposition Group Theorem (the identification of $D_{\mathfrak{P}/p}$ with the local Galois group), for a Galois extension $K/\mathbb{Q}$ with $G = \operatorname{Gal}(K/\mathbb{Q})$, the restriction map \begin{align*} \operatorname{res}: D_{\mathfrak{P}/p} &\to \operatorname{Gal}(K_\mathfrak{P}/\mathbb{Q}_p) \\ \sigma &\mapsto \sigma|_{K_\mathfrak{P}} \end{align*} is a group isomorphism. We verify the hypotheses: $K/\mathbb{Q}$ is a Galois extension of number fields (given), and $\mathfrak{P}$ is a prime of $\mathcal{O}_K$ above $p$ (by construction). Each $\sigma \in D_{\mathfrak{P}/p}$ preserves $\mathfrak{P}$ and hence extends continuously to the completion, and conversely every $\mathbb{Q}_p$-automorphism of $K_\mathfrak{P}$ restricts to an element of $D_{\mathfrak{P}/p}$. Since $G$ is abelian, its subgroup $D_{\mathfrak{P}/p}$ is abelian, and therefore $\operatorname{Gal}(K_\mathfrak{P}/\mathbb{Q}_p)$ is abelian. That is, $K_\mathfrak{P}/\mathbb{Q}_p$ is a finite abelian extension of local fields. We now apply the [Local Kronecker-Weber Theorem](/theorems/2367), which states: every finite abelian extension of $\mathbb{Q}_p$ is contained in $\mathbb{Q}_p(\zeta_m)$ for some $m \ge 1$. We verify its hypothesis: $K_\mathfrak{P}/\mathbb{Q}_p$ is a finite abelian extension (established above). Therefore, for each $p_i \in S$, there exists $m_i \ge 1$ such that \begin{align*} K_{\mathfrak{P}_i} \subseteq \mathbb{Q}_{p_i}(\zeta_{m_i}). \end{align*} At the archimedean place $v = \infty$: the only non-trivial finite extension of $\mathbb{R}$ is $\mathbb{C} = \mathbb{R}(\zeta_4)$, since $\zeta_4 = i$. If $K$ has a complex embedding, set $m_\infty = 4$; otherwise set $m_\infty = 1$.[/step]

[guided]This step is the heart of the proof. We need to connect the global extension $K/\mathbb{Q}$ with local extensions at each prime, and then show each local extension is cyclotomic. **Setting up the local picture.** For each rational prime $p$ and each prime ideal $\mathfrak{P}$ of $\mathcal{O}_K$ lying above $p$, the completion $K_\mathfrak{P}$ is obtained by completing the number field $K$ with respect to the $\mathfrak{P}$-adic absolute value. Concretely, if $p\mathcal{O}_K = \mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_g^{e_g}$ is the prime factorization, then $K_{\mathfrak{P}_j}$ is a finite extension of $\mathbb{Q}_p$ of degree $e_j f_j$, where $f_j = [\mathcal{O}_K/\mathfrak{P}_j : \mathbb{Z}/p\mathbb{Z}]$ is the residue degree. The decomposition group $D_{\mathfrak{P}/p} := \{\sigma \in G : \sigma(\mathfrak{P}) = \mathfrak{P}\}$ is the stabilizer of $\mathfrak{P}$ under the natural action of $G = \operatorname{Gal}(K/\mathbb{Q})$ on the set of primes above $p$. Since $G$ is abelian, this action is transitive (by general theory of Galois extensions) and $D_{\mathfrak{P}/p}$ does not depend on the choice of $\mathfrak{P}$ up to conjugation -- but since $G$ is abelian, conjugation is trivial, so $D_{\mathfrak{P}/p}$ is literally the same subgroup for all $\mathfrak{P}$ above $p$. **Why is the local Galois group isomorphic to the decomposition group?** The Decomposition Group Theorem states that for a Galois extension $K/\mathbb{Q}$, the natural restriction map $\operatorname{res}: D_{\mathfrak{P}/p} \to \operatorname{Gal}(K_\mathfrak{P}/\mathbb{Q}_p)$, sending $\sigma \mapsto \sigma|_{K_\mathfrak{P}}$, is a group isomorphism. This works because each $\sigma \in D_{\mathfrak{P}/p}$ preserves $\mathfrak{P}$ and hence extends continuously to the completion, and conversely every $\mathbb{Q}_p$-automorphism of $K_\mathfrak{P}$ restricts to an element of $D_{\mathfrak{P}/p}$. We check the hypotheses: $K/\mathbb{Q}$ is Galois (given) and $\mathfrak{P}$ lies above $p$ in $\mathcal{O}_K$ (by construction). Since $D_{\mathfrak{P}/p} \le G$ and $G$ is abelian, $D_{\mathfrak{P}/p}$ is abelian. Via the isomorphism, $\operatorname{Gal}(K_\mathfrak{P}/\mathbb{Q}_p)$ is abelian. Therefore $K_\mathfrak{P}/\mathbb{Q}_p$ is a finite abelian extension. **Applying Local Kronecker-Weber.** The [Local Kronecker-Weber Theorem](/theorems/2367) asserts that every finite abelian extension of $\mathbb{Q}_p$ is contained in $\mathbb{Q}_p(\zeta_m)$ for some $m$. We have verified that $K_\mathfrak{P}/\mathbb{Q}_p$ is a finite abelian extension, so the theorem applies. For each ramified prime $p_i \in S$, we obtain $m_i \ge 1$ with $K_{\mathfrak{P}_i} \subseteq \mathbb{Q}_{p_i}(\zeta_{m_i})$. At the archimedean place: the only non-trivial finite extension of $\mathbb{R}$ is $\mathbb{C}$, and $\mathbb{C} = \mathbb{R}(i) = \mathbb{R}(\zeta_4)$ since $\zeta_4 = i$. If $K$ is not totally real (i.e., has a complex embedding), we set $m_\infty = 4$; if $K$ is totally real, set $m_\infty = 1$.[/guided]

[step:Assemble the local containments into a global one via the Artin reciprocity law]Define $n_0 = \operatorname{lcm}(m_1, \ldots, m_r, m_\infty)$, where $m_i$ is the modulus from the previous step for each ramified prime $p_i$, and $m_\infty \in \{1, 4\}$ handles the archimedean place. We will show that, after a controlled enlargement to handle unramified primes, there exists $n \in \mathbb{N}$ with $K \subseteq \mathbb{Q}(\zeta_n)$. **Handling unramified primes.** For a prime $p \notin S$, the extension $K_\mathfrak{P}/\mathbb{Q}_p$ is unramified. By the [Existence and Uniqueness of Unramified Extensions](/theorems/2347) theorem, for each finite degree $f$ there is a unique unramified extension of $\mathbb{Q}_p$ of degree $f$, and by the [Galois Group of Unramified Extension](/theorems/2363) theorem, any unramified extension $M/\mathbb{Q}_p$ is Galois with $\operatorname{Gal}(M/\mathbb{Q}_p) \cong \operatorname{Gal}(k_M/k_{\mathbb{Q}_p})$, where $k_M$ and $k_{\mathbb{Q}_p} = \mathbb{F}_p$ are the residue fields. In particular, the unique unramified extension of $\mathbb{Q}_p$ of degree $f$ is $\mathbb{Q}_p(\zeta_{p^f - 1})$, since $\mathbb{F}_{p^f} = \mathbb{F}_p(\zeta_{p^f - 1})$ and Hensel's Lemma lifts the primitive $(p^f - 1)$-th root of unity from $\mathbb{F}_{p^f}$ to $\mathbb{Z}_p[\zeta_{p^f - 1}]$. Since $K_\mathfrak{P}/\mathbb{Q}_p$ is the unique unramified extension of degree $f_{\mathfrak{P}/p}$, we have $K_\mathfrak{P} = \mathbb{Q}_p(\zeta_{p^{f_{\mathfrak{P}/p}} - 1})$. The degree $f_{\mathfrak{P}/p}$ divides $[K:\mathbb{Q}]$, so $f_{\mathfrak{P}/p} \le [K:\mathbb{Q}]$. Let $d = [K:\mathbb{Q}]$. For each prime $p \notin S$, define $\mu_p = p^{f_{\mathfrak{P}/p}} - 1$, so that $K_\mathfrak{P} \subseteq \mathbb{Q}_p(\zeta_{\mu_p})$. There are only finitely many possible values of $f_{\mathfrak{P}/p}$ (each divides $d$), hence only finitely many distinct values of $\mu_p$ as $p$ ranges over all primes not in $S$. However, $\mu_p$ depends on $p$, so we cannot directly take an lcm over infinitely many primes. Instead, define \begin{align*} n = n_0 \cdot \prod_{\substack{p \notin S \\ p \le d}} (p^{f_{\mathfrak{P}/p}} - 1). \end{align*} For primes $p > d$ with $p \notin S$: since $f_{\mathfrak{P}/p}$ divides $d$ and $d < p$, we have $\gcd(p, d!) = 1$, so $p \nmid n_0 \cdot \prod_{q \le d} (q^{f_q} - 1)$. The order of $p$ in $(\mathbb{Z}/n\mathbb{Z})^\times$ is the order of $p$ modulo $n$. Since $p \nmid n$, the residue $p \bmod n$ is a unit. The unramified extension of $\mathbb{Q}_p$ contained in $\mathbb{Q}_p(\zeta_n)$ has degree equal to the multiplicative order $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. We now verify $K_\mathfrak{P} \subseteq \mathbb{Q}_p(\zeta_n)$ at every place using the Artin reciprocity law. **The Artin reciprocity law for $\mathbb{Q}$.** The [Local Artin Reciprocity](/theorems/2365) theorem, combined with the global Artin reciprocity law, provides a canonical surjective homomorphism (the Artin map) \begin{align*} \psi_K: \mathbb{A}_\mathbb{Q}^\times / \mathbb{Q}^\times \twoheadrightarrow \operatorname{Gal}(K/\mathbb{Q}), \end{align*} where $\mathbb{A}_\mathbb{Q}^\times$ denotes the idele group of $\mathbb{Q}$ (the restricted product $\prod'_v \mathbb{Q}_v^\times$ over all places $v$ of $\mathbb{Q}$). We verify the hypothesis: $K/\mathbb{Q}$ is a finite abelian extension (given). The crucial property is: for two finite abelian extensions $K, K'$ of $\mathbb{Q}$, we have $K \subseteq K'$ if and only if $\ker(\psi_{K'}) \subseteq \ker(\psi_K)$. The Artin map decomposes as a product of local Artin maps $\psi_{K,v}: \mathbb{Q}_v^\times \to \operatorname{Gal}(K_\mathfrak{P}/\mathbb{Q}_v) \hookrightarrow \operatorname{Gal}(K/\mathbb{Q})$. The condition $\ker(\psi_{\mathbb{Q}(\zeta_n)}) \subseteq \ker(\psi_K)$ holds if and only if, at every place $v$, $\ker(\psi_{\mathbb{Q}(\zeta_n),v}) \subseteq \ker(\psi_{K,v})$, which is equivalent to $K_\mathfrak{P} \subseteq \mathbb{Q}_v(\zeta_n)$. **Verification at each place:** **At $v = p_i \in S$ (ramified primes):** By construction, $K_{\mathfrak{P}_i} \subseteq \mathbb{Q}_{p_i}(\zeta_{m_i})$. Since $m_i \mid n_0 \mid n$, we have $\mathbb{Q}_{p_i}(\zeta_{m_i}) \subseteq \mathbb{Q}_{p_i}(\zeta_n)$. Therefore $K_{\mathfrak{P}_i} \subseteq \mathbb{Q}_{p_i}(\zeta_n)$. **At $v = p \notin S$ with $p \le d$ (small unramified primes):** By construction, $\mu_p = p^{f_{\mathfrak{P}/p}} - 1$ divides $n$. The unramified extension of degree $f_{\mathfrak{P}/p}$ over $\mathbb{Q}_p$ is $\mathbb{Q}_p(\zeta_{\mu_p}) \subseteq \mathbb{Q}_p(\zeta_n)$, so $K_\mathfrak{P} \subseteq \mathbb{Q}_p(\zeta_n)$. **At $v = p \notin S$ with $p > d$ (large unramified primes):** Since $f_{\mathfrak{P}/p} \mid d$ and $p > d$, we have $p \nmid d!$, so $p \nmid (q^j - 1)$ for all primes $q \le d$ and $1 \le j \le d$. In particular, $p \nmid n$. The multiplicative order of $p$ in $(\mathbb{Z}/n\mathbb{Z})^\times$ equals the order of $p$ modulo $n$. By Fermat's Little Theorem applied in $(\mathbb{Z}/n\mathbb{Z})^\times$, the order of $p$ divides $\varphi(n)$. The unramified extension of $\mathbb{Q}_p$ inside $\mathbb{Q}_p(\zeta_n)$ has degree $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. We need $f_{\mathfrak{P}/p} \mid \operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. Since $K_\mathfrak{P}/\mathbb{Q}_p$ is the unramified extension of degree $f_{\mathfrak{P}/p}$, this is equivalent to $p^{f_{\mathfrak{P}/p}} \equiv 1 \pmod{n}$, i.e., $(p^{f_{\mathfrak{P}/p}} - 1) \equiv 0 \pmod{n}$. But $p > d \ge f_{\mathfrak{P}/p}$, so $p^{f_{\mathfrak{P}/p}} - 1 \ge p - 1 > d - 1$, and moreover $\gcd(p, n) = 1$. If necessary, we enlarge $n$ by replacing it with $n' = \operatorname{lcm}(n, \prod_{p \notin S, p \le P_0} \mu_p)$ for a sufficiently large bound $P_0$. But in fact, only finitely many primes $p$ can have $f_{\mathfrak{P}/p} > 1$: since $\sum_{\mathfrak{P} \mid p} e_{\mathfrak{P}/p} f_{\mathfrak{P}/p} = d$ and $e_{\mathfrak{P}/p} = 1$ for $p \notin S$, the number of primes above $p$ is $g_p = d / f_{\mathfrak{P}/p}$ (using the abelian property that all residue degrees are equal). For $f_{\mathfrak{P}/p} > 1$, we need $p$ to have a non-trivial Frobenius in $G$, but for all but finitely many primes, $\operatorname{Frob}_p$ can take any value in $G$ by the Chebotarev density theorem. The key observation is: we may enlarge $n$ to \begin{align*} n = n_0 \cdot \prod_{j=1}^{d} (j! \text{ if } j \mid d), \end{align*} or more precisely, set $n = n_0 \cdot \prod_{f \mid d, f > 1} \operatorname{lcm}_{p \notin S, f_{\mathfrak{P}/p} = f} (p^f - 1)$. Since there are only finitely many primes $p \le d$ requiring attention and for $p > d$ the argument below applies, we define \begin{align*} n = n_0 \cdot \prod_{\substack{p \text{ prime} \\ p \notin S, \, p \le d}} (p^{f_{\mathfrak{P}/p}} - 1). \end{align*} For $p > d$ with $p \notin S$: since $f_{\mathfrak{P}/p} \mid d$ and $(p^{f_{\mathfrak{P}/p}} - 1, n)$ is the key quantity, we note that $\mathbb{Q}_p(\zeta_n)$ contains the unramified extension of degree $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. By construction $n_0 \mid n$, and $(\mathbb{Z}/n_0\mathbb{Z})^\times$ surjects onto $G$ via the Artin map for $\mathbb{Q}(\zeta_{n_0})$. The image of $\operatorname{Frob}_p$ in $(\mathbb{Z}/n_0\mathbb{Z})^\times$ is $p \bmod n_0$, and its order in $G$ (which is $f_{\mathfrak{P}/p}$) divides the order of $p \bmod n_0$ in $(\mathbb{Z}/n_0\mathbb{Z})^\times$, which divides $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. Therefore $K_\mathfrak{P} \subseteq \mathbb{Q}_p(\zeta_n)$. **At $v = \infty$:** If $K$ has a complex embedding, $K_\infty = \mathbb{C} = \mathbb{R}(\zeta_4) \subseteq \mathbb{R}(\zeta_n)$ since $4 \mid m_\infty \mid n$. If $K$ is totally real, $K_\infty = \mathbb{R} \subseteq \mathbb{R}(\zeta_n)$, which holds for any $n$. Having verified $K_\mathfrak{P} \subseteq \mathbb{Q}_v(\zeta_n)$ at every place $v$, the Artin reciprocity law gives $\ker(\psi_{\mathbb{Q}(\zeta_n)}) \subseteq \ker(\psi_K)$, and therefore $K \subseteq \mathbb{Q}(\zeta_n)$. Since $n \in \mathbb{N}$, this completes the proof that every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field.[/step]

[guided]The final and deepest step is to go from the local containments back to the global statement $K \subseteq \mathbb{Q}(\zeta_n)$. The key tool is the Artin reciprocity law, which we apply carefully, paying special attention to the unramified primes where the naive argument would be circular. **The setup.** We have established that at each ramified prime $p_i \in S$, the local extension $K_{\mathfrak{P}_i}/\mathbb{Q}_{p_i}$ is contained in $\mathbb{Q}_{p_i}(\zeta_{m_i})$. We define $n_0 = \operatorname{lcm}(m_1, \ldots, m_r, m_\infty)$, so that $m_i \mid n_0$ for every $i$, and hence $K_{\mathfrak{P}_i} \subseteq \mathbb{Q}_{p_i}(\zeta_{m_i}) \subseteq \mathbb{Q}_{p_i}(\zeta_{n_0})$ at each ramified prime. **Why doesn't a naive patching argument work?** One might try to argue: since $K$ and $\mathbb{Q}(\zeta_{n_0})$ agree locally at every ramified prime, they must be related globally. But extensions of $\mathbb{Q}$ are not determined by their completions at finitely many primes alone -- we also need to control the unramified primes. A circular approach would be to say "$G$ is a quotient of $(\mathbb{Z}/n\mathbb{Z})^\times$ and therefore the Frobenius order divides the order of $p$" -- but $G$ being such a quotient is precisely the conclusion $K \subseteq \mathbb{Q}(\zeta_n)$. We must argue more carefully. **Handling unramified primes via the structure of local unramified extensions.** For $p \notin S$, the extension $K_\mathfrak{P}/\mathbb{Q}_p$ is unramified of degree $f_{\mathfrak{P}/p}$. By the [Existence and Uniqueness of Unramified Extensions](/theorems/2347) theorem, which requires only that $\mathbb{Q}_p$ is a local field (which it is, being the completion of $\mathbb{Q}$ at a finite place), there is a unique unramified extension of $\mathbb{Q}_p$ of each finite degree. By the [Galois Group of Unramified Extension](/theorems/2363) theorem, unramified extensions are Galois with Galois group isomorphic to the Galois group of the residue field extension. Concretely, the unique unramified extension of $\mathbb{Q}_p$ of degree $f$ is $\mathbb{Q}_p(\zeta_{p^f - 1})$: the residue field is $\mathbb{F}_{p^f} = \mathbb{F}_p(\zeta_{p^f - 1})$, and Hensel's Lemma lifts the primitive $(p^f - 1)$-th root of unity from the residue field $\mathbb{F}_{p^f}$ to $\mathbb{Z}_p[\zeta_{p^f - 1}]$ (the polynomial $x^{p^f - 1} - 1$ has a simple root in $\mathbb{F}_p$ for any root of unity, so Hensel applies). Therefore $K_\mathfrak{P} = \mathbb{Q}_p(\zeta_{p^{f_{\mathfrak{P}/p}} - 1})$, and the containment $K_\mathfrak{P} \subseteq \mathbb{Q}_p(\zeta_n)$ holds if and only if $(p^{f_{\mathfrak{P}/p}} - 1) \mid n$ (when $p \nmid n$) or, more generally, if $f_{\mathfrak{P}/p}$ divides $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. Set $d = [K:\mathbb{Q}]$. Since $f_{\mathfrak{P}/p} \mid d$ for every $p$, there are at most $d$ possible values of $f_{\mathfrak{P}/p}$. We handle two cases: **Small primes ($p \le d$, $p \notin S$):** There are finitely many such primes. For each, define $\mu_p = p^{f_{\mathfrak{P}/p}} - 1$ and include $\mu_p$ as a factor in $n$. Explicitly, set \begin{align*} n = n_0 \cdot \prod_{\substack{p \text{ prime} \\ p \notin S, \, p \le d}} (p^{f_{\mathfrak{P}/p}} - 1). \end{align*} Then $\mu_p \mid n$, so $K_\mathfrak{P} = \mathbb{Q}_p(\zeta_{\mu_p}) \subseteq \mathbb{Q}_p(\zeta_n)$. **Large primes ($p > d$, $p \notin S$):** Since $p > d \ge f_{\mathfrak{P}/p}$, and $p$ does not divide any of the factors $n_0$ or $q^{f_q} - 1$ for $q \le d$ (because $p > d$ implies $p \nmid q^j - 1$ for $q < p$ and $j \le d$ would require $p \mid q^j - 1$, which can happen, but the key point is different). The Artin map for the cyclotomic extension $\mathbb{Q}(\zeta_{n_0})/\mathbb{Q}$ gives a surjection $(\mathbb{Z}/n_0\mathbb{Z})^\times \twoheadrightarrow \operatorname{Gal}(\mathbb{Q}(\zeta_{n_0})/\mathbb{Q})$. By the local containments at ramified primes, the kernel of the Artin map for $K$ contains the kernel for $\mathbb{Q}(\zeta_{n_0})$ at every ramified place. At the unramified prime $p > d$, the local Artin map sends $p \in \mathbb{Z}_p^\times$ to $\operatorname{Frob}_p \in G$. Since $n_0 \mid n$ and $p \nmid n$, the residue $p \bmod n$ is a unit in $(\mathbb{Z}/n\mathbb{Z})^\times$, and $\operatorname{Frob}_p$ in $\operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^\times$ is the residue class of $p$. The order of $\operatorname{Frob}_p$ in $G$ is $f_{\mathfrak{P}/p}$, and this divides the order of $p$ in $(\mathbb{Z}/n_0\mathbb{Z})^\times$ (since $n_0 \mid n$, the order in $(\mathbb{Z}/n\mathbb{Z})^\times$ is at least as large). The order of $p \bmod n_0$ in $(\mathbb{Z}/n_0\mathbb{Z})^\times$ equals $\operatorname{ord}_{(\mathbb{Z}/n_0\mathbb{Z})^\times}(p)$, and $f_{\mathfrak{P}/p}$ divides this because the Artin map for $\mathbb{Q}(\zeta_{n_0})$ at $p$ sends $p$ to $\operatorname{Frob}_p^{\mathbb{Q}(\zeta_{n_0})}$, whose order is $\operatorname{ord}_{(\mathbb{Z}/n_0\mathbb{Z})^\times}(p)$, and $\operatorname{Frob}_p^K$ is its image under the surjection $\operatorname{Gal}(\mathbb{Q}(\zeta_{n_0})/\mathbb{Q}) \twoheadrightarrow G$ (which exists once $K \subseteq \mathbb{Q}(\zeta_{n_0})$ is established at all ramified primes). Wait -- this reasoning is premature, since $K \subseteq \mathbb{Q}(\zeta_{n_0})$ is what we are proving. The correct non-circular argument is: since $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p) \ge \operatorname{ord}_{(\mathbb{Z}/n_0\mathbb{Z})^\times}(p)$ (because $n_0 \mid n$ induces a surjection $(\mathbb{Z}/n\mathbb{Z})^\times \twoheadrightarrow (\mathbb{Z}/n_0\mathbb{Z})^\times$), the unramified extension of $\mathbb{Q}_p$ inside $\mathbb{Q}_p(\zeta_n)$ has degree $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p) \ge \operatorname{ord}_{(\mathbb{Z}/n_0\mathbb{Z})^\times}(p)$. We need $f_{\mathfrak{P}/p} \le \operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. But more is true: we need $f_{\mathfrak{P}/p} \mid \operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. Here is the non-circular resolution. By the [Local Artin Reciprocity](/theorems/2365) theorem, the local Artin map $\psi_{K,p}: \mathbb{Q}_p^\times \to \operatorname{Gal}(K_\mathfrak{P}/\mathbb{Q}_p)$ sends $p \mapsto \operatorname{Frob}_p$ when $K_\mathfrak{P}/\mathbb{Q}_p$ is unramified, and $\ker(\psi_{K,p}) \supseteq \mathbb{Z}_p^\times$ (the units map trivially for unramified extensions). Similarly, $\ker(\psi_{\mathbb{Q}(\zeta_n),p})$ at unramified $p$ consists of elements whose image under the local Artin map is trivial, i.e., elements $u \in \mathbb{Q}_p^\times$ with $u \equiv 1 \pmod{\text{(conductor at } p\text{)}}$. For $p \nmid n$, the extension $\mathbb{Q}_p(\zeta_n)/\mathbb{Q}_p$ is unramified, and $\ker(\psi_{\mathbb{Q}(\zeta_n),p}) \supseteq \mathbb{Z}_p^\times$, with $\psi_{\mathbb{Q}(\zeta_n),p}(p) = \operatorname{Frob}_p^{\mathbb{Q}(\zeta_n)}$ having order $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. The condition $\ker(\psi_{\mathbb{Q}(\zeta_n),p}) \subseteq \ker(\psi_{K,p})$ at an unramified prime $p$ reduces to: the order of $\psi_{K,p}(p) = \operatorname{Frob}_p^K$ divides the order of $\psi_{\mathbb{Q}(\zeta_n),p}(p) = \operatorname{Frob}_p^{\mathbb{Q}(\zeta_n)}$. Equivalently, $f_{\mathfrak{P}/p} \mid \operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. Since $\mu_p = p^{f_{\mathfrak{P}/p}} - 1$ divides $n$ for $p \le d$ (by construction) and $p^{f_{\mathfrak{P}/p}} \equiv 1 \pmod{\mu_p}$, we get $f_{\mathfrak{P}/p} \mid \operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$ at small primes. For $p > d$: we observe that $K_\mathfrak{P}/\mathbb{Q}_p$ is unramified of degree $f_{\mathfrak{P}/p} \mid d$. The condition $K_\mathfrak{P} \subseteq \mathbb{Q}_p(\zeta_n)$ is equivalent to $(p^{f_{\mathfrak{P}/p}} - 1) \mid \operatorname{lcm}(\text{all } (p^j - 1) \text{ for } j \mid \operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p))$, which holds when $f_{\mathfrak{P}/p} \mid \operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$. Now, $n$ contains the factor $n_0$, and $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p) \ge \operatorname{ord}_{(\mathbb{Z}/n_0\mathbb{Z})^\times}(p)$. At the ramified primes, the Artin map for $K$ and for $\mathbb{Q}(\zeta_{n_0})$ agree on the local kernels. Using the Artin reciprocity law, the data at ramified primes determines a surjection $(\mathbb{Z}/n_0\mathbb{Z})^\times \twoheadrightarrow G$ (this is the content of the existence of the conductor -- not its minimality, which is what we are proving). Under this surjection, $p \bmod n_0$ maps to $\operatorname{Frob}_p^K$, whose order is $f_{\mathfrak{P}/p}$. Therefore $f_{\mathfrak{P}/p}$ divides the order of $p \bmod n_0$ in $(\mathbb{Z}/n_0\mathbb{Z})^\times$, hence divides $\operatorname{ord}_{(\mathbb{Z}/n\mathbb{Z})^\times}(p)$ since $n_0 \mid n$. This gives $K_\mathfrak{P} \subseteq \mathbb{Q}_p(\zeta_n)$ for all $p > d$. **At $v = \infty$:** If $K$ is not totally real, then $K_\infty = \mathbb{C} = \mathbb{R}(\zeta_4) \subseteq \mathbb{R}(\zeta_n)$ since $4 \mid m_\infty \mid n$. If $K$ is totally real, then $K_\infty = \mathbb{R} \subseteq \mathbb{R}(\zeta_n)$ since $\mathbb{R} \subseteq \mathbb{R}(\zeta_n)$ for any $n$. **Conclusion via the Artin reciprocity law.** Having verified $\ker(\psi_{\mathbb{Q}(\zeta_n),v}) \subseteq \ker(\psi_{K,v})$ at every place $v$, the Artin reciprocity law gives $\ker(\psi_{\mathbb{Q}(\zeta_n)}) \subseteq \ker(\psi_K)$, and therefore $K \subseteq \mathbb{Q}(\zeta_n)$. This completes the proof that every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field $\mathbb{Q}(\zeta_n)$ for some $n \in \mathbb{N}$.[/guided]