Roots of unity inside ring class fields of an imaginary quadratic field $K$ always contained in $K(\mu_{12})$?

Question

I read online (in https://mathoverflow.net/questions/184347/intersection-of-a-ring-class-field-of-a-quadratic-field-k-with-the-cyclotomic-ex) that an (apparently not so trivial) exercise in class field theory shows that the roots of unity contained in ring class fields (of any conductor) on an imaginary quadratic field $K$ are always contained in $K(\mu_{12})$, where $\mu_{12}$ is the group of 12-th roots of unity.

Unfortunately, I can't find any proof or reference of this fact: can someone help me?

Answer 1

Not an expert of those things so please check that I'm not confused with something, and I can only show that the roots of unity of $K_O$ are contained in $\langle\zeta_{24}\rangle$.

We have an order $O=\Bbb{Z}+dO_K$.

The Artin map of the ring class field $K_O/K$ is $$(I,K_O/K)= [I\cap O]\ \in Cl(O)$$

For $a\in \Bbb{Z},\gcd(a,d)=1$ then $aO_K\cap O=aO$.

Proof: take $b\in \Bbb{Z},ab=1\bmod n$, if $av=c+dw\in aO_K\cap O$ with $c\in \Bbb{Z},v,w\in O_K$ then $v=abv=bc\in (O_K/dO_K)^\times$ and hence $v\in O$.

Fix $n\in \Bbb{Z},n\nmid 24$. The Artin map of $K(\zeta_n)/K$ is $$(I,K(\zeta_n)/K)=N(I)\in \Bbb{Z}/n\Bbb{Z}^\times$$

For some $a\in \Bbb{Z},\gcd(a,dn)=1$ we have $a^2 \ne 1\bmod n$ so that $$(aO_K,K(\zeta_n)/K)=N(aO_K)=a^2\ne 1\in \Bbb{Z}/n\Bbb{Z}^\times$$ whereas

$$(aO_K,K_O/K)=[aO_K\cap O]=[aO]=1\in Cl(O)$$

whence $\zeta_n\not \in K_O$.