Looking for examples of abelian extensions of $\mathbb{Q}$ satisfying a certain condition

Question

The Kronecker-Weber Theorem says that every abelian extension of $\mathbb{Q}$ is contained in some cyclotomic extension. One approach to prove this is via higher ramification groups. In this approach, there is a crucial step, which says

Let $K$ be an abelain extension of $\mathbb{Q}$ such that $[K:\mathbb{Q}]=p^m$ and $p$ is the only prime of $\mathbb{Z}$ which ramifies in $K$. To prove the Kronecker-Weber theorem, it is enough to prove that any such $K$ is contained in a cyclotomic extension.

I am looking for some examples such a $K$. Any help ?

Answer 1

$\Bbb Q(i), \Bbb Q(\sqrt 2), \Bbb Q(\cos(2\pi/9))$ are some examples among many others.

Answer 2

Assume $p$ odd for simplificaion. For any integer $n\ge 1$, consider the cyclomic extension $L_n = \mathbf Q(\zeta_n)$ obtained by adding a primitive $n$-th root of $1$. It is classically known that $Gal(L_1/K)$ is cyclic of order dividing $(p-1)$ and $Gal(L_{m+1} /L_1)$ is cyclic of order $p^m$. Moreover, since $p$ and $(p-1)$ are coprime, the abelian group $Gal(L_{m+1}/\mathbf Q)$ is isomorphic to the direct product of $Gal(L_1/\mathbf Q)$ and $Gal(L_{m+1} /L_1)$ (this is pure group theory). It follows that there exists an extension $K_{m+1}/\mathbf Q$ such that $L_{m+1}$ is the compositum of $L_1$ and $K_{m+1}$, and $Gal(K_{m+1} /K_1)$ is cyclic of order $p^m$. Moreover the general theory of ramification in cyclotomic extensions (see e.g. Marcus' "Number Fields") tells us that $p$ is the only prime of $\mathbf Z$ which ramifies in $L_{m+1}$ (and it is totally ramified). The field $K_{m+1}$ gives the example you wanted.