Generalization of Kronecker–Weber theorem: is an abelian extension always cyclotomic?

Question

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!

Answer 1

This is false for every number field $K$ other than $\mathbf Q$. The general proof uses ideas related to class field theory.

As a concrete example, consider the quadratic extension $\mathbf Q(\sqrt[4]{2})/\mathbf Q(\sqrt{2})$. Since $\mathbf Q(\sqrt{2})$ is inside a cyclotomic extension of $\mathbf Q$, namely it is in $\mathbf Q(\zeta_8)$, any cyclotomic extension of $\mathbf Q(\sqrt{2})$ is inside a cyclotomic extension of $\mathbf Q$. Therefore any cyclotomic extension of $\mathbf Q(\sqrt{2})$ is an abelian extension of $\mathbf Q$. Since $\mathbf Q(\sqrt[4]{2})$ is not abelian (or even Galois) over $\mathbf Q$, it is not contained in a cyclotomic extension of $\mathbf Q(\sqrt{2})$.

Many more counterexamples can be made by replacing $\mathbf Q(\sqrt{2})$ with other abelian extensions of $\mathbf Q$.

Answer 2

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.