Keith Conrad's History of Class Field Theory notes has a comment that given any number field $K\neq \mathbb{Q}$, then $K$ has an abelian extension which is not contained in $K(\zeta_\infty)$, so the Kronecker-Weber theorem for $K$ is false. I haven't been able to find a proof of this. I assume that we have to look at the idelic formulation of global class field theory to find it. We have the following diagram of extensions (I can't figure out how to make vertical lines):
$K$ —— $K(\zeta_\infty)$ —— $K^{ab}$
$\mathbb{Q}$ —— $\mathbb{Q}^{ab}$
and functoriality of idelic CFT, then gives the diagram:
$C_K$ ——> $\textrm{Gal}(K^{ab}/K)$
Norm ….. Restriction
$C_\mathbb{Q}$ ——> $\textrm{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$
Since any automorphism of a cyclotomic extension is uniquely determined by where it maps the different roots of unity, the restriction map would have to be injective if $K(\zeta_\infty)=K^{ab}$. If not, then the map clearly can't be injective, since if there's some $L/K$ that is abelian and not contained in a cyclotomic extension, then we can pick a non-trivial automorphism of $L$ that fixes the maximal cyclotomic subextension.
My question essentially has two parts:
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Is it enough to show that the norm map is not injective? What's confusing me is that the horizontal maps are not isomorphisms.
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How would one find the kernel of the norm map? I seem to get lost trying to do this computation.
EDIT: I just realized I've been stupid since I don't need to know the whole kernel just that it's noninjective… Pick any splitting prime $(p)=\mathfrak{p}_1\cdots \mathfrak{p}_n$, so
$$K_{\mathfrak{p}_i}=\mathbb{Q}_p.$$
Now just pick an element $(a_i)\in C_K$ where the element sitting in $K_{\mathfrak{p}_1}$ is $2$ and the element sitting in $\mathfrak{p}_2$ is $1/2$ and the rest are $1$. This should give norm $1$ if I understand the definition correctly?
Best Answer
Edited following comments of Wanax:
This is just an amplification of Wanax's comments: All you need to do is to pick a prime $p$ of $Q$ which splits into (more than two) distinct primes $\mathfrak p_1$ and $\mathfrak p_2$ in $K$ and which is unramified in $K$. There are infinite number of such primes if $K$ is distinct from $Q$. Pick an element $\alpha$ of $K$ which is divisible by $\mathfrak p_1$ but not $\mathfrak p_2$. Then the quadratic extension $L= K(\sqrt{\alpha})$ is an abelian extension of $K$ and this $L$ gives the extension you want. Because if $L$ were obtained by adjoining a root of unity to $K$, both primes $\mathfrak p_1$ and $\mathfrak p_2$ would have to be ramified in $L$ but our choice ensures that only $\mathfrak p_1$ is ramified in $L$.