For each prime

\(p\) over which

\(K\) ramifies, pick a prime

\(\gothp\) of

\(K\) over

\(p\text{;}\) by local Kronecker-Weber (

Theorem 1.1.5),

\(K_{\gothp} \subseteq \QQ_p(\zeta_{n_p})\) for some positive integer

\(n_p\text{.}\) Let

\(p^{e_p}\) be the largest power of

\(p\) dividing

\(n_p\text{,}\) and put

\(n = \prod_p p^{e_p}\text{.}\) (This is a finite product since only finitely many primes ramify in

\(K\text{.}\))

Write \(L = K(\zeta_n)\text{;}\) we will prove that \(K \subseteq \QQ(\zeta_n)\) by proving that \(L = \QQ(\zeta_n)\text{.}\) Form the completion \(L_\gothq\) for some prime \(\gothq\) over \(p\text{;}\) it is contained in \(\QQ_p(\zeta_{\lcm(n,n_p)})\text{.}\) Let \(I_p\) be the inertia group of \(p\) in \(L\text{;}\) the fixed fixed \(U\) of \(I_p\) on \(L_{\gothq}\) is the maximal unramified subextension of \(L_\gothq\text{.}\) Since \(\QQ_p(\zeta_e)\) is unramified over \(\QQ_p\) for any positive integer \(e\) coprime to \(p\text{,}\) we have \(L_{\gothq} = U(\zeta_{p^{e_p}})\) and so \(I_p \cong \Gal(L_\gothq/U) \subseteq (\ZZ/p^{e_p}\ZZ)^*\text{.}\) Let \(I\) be the group generated by all of the \(I_p\text{;}\) then

\begin{equation*}
|I| \leq \prod |I_p| \leq \prod \phi(p^{e_p}) = \phi(n) = [\QQ(\zeta_n):\QQ].
\end{equation*}

On the other hand, the fixed field of \(I\) is an everywhere unramified extension of \(\QQ\text{,}\) which can only be \(\QQ\) itself by Minkowski’s theorem. That is, \(I = \Gal(L/\QQ)\text{.}\) But then

\begin{equation*}
[L:\QQ] = |I| \leq [\QQ(\zeta_n):\QQ],
\end{equation*}

and \(\QQ(\zeta_n) \subseteq L\text{,}\) so we must have \(\QQ(\zeta_n) = L\) and \(K \subseteq \QQ(\zeta_n)\text{,}\) as desired.