Generalization of Kronecker–Weber theorem: is an abelian extension always cyclotomic? It is well-known that any abelian extension of $\Bbb Q$ is contained within some cyclotomic field $\Bbb Q(\zeta_n)$.  My question is about the more general statement:

If $L/K$ is an abelian extension, do we have $L \subset K(\zeta_n)$, where $\zeta_n \in \overline K$ is a root of $X^n-1$ (for some $n \in \Bbb N$) ?

For instance, any abelian extension of $\Bbb Q_p$ is contained within some cyclotomic field $\Bbb Q_p(\zeta_n)$. Morover, it holds when $K$ is a finite field (the Galois groups are cyclic, and any non zero element is a root of unity).
I believe that the statement is wrong, but I wasn't able to come up with a counterexample.
Thank you very much!

No.
Consider the extension $\mathbb{Q}(t^{1/2})$ of the field $\mathbb{Q}(t)$.  It is an extension of degree $2$, and is thus abelian.  However, adding any root of unity $\zeta$ to $\mathbb{Q}(t)$ will give you the field $(\mathbb{Q}(\zeta))(t)$, in which $t$ has no square root.
