Relation between the genus number and the ambiguous class number

Question

It is well known that for $K/F$ a finite "cyclic" extension of number fields, we have $g_{K/F}=a_{K/F}$, where $g_{K/F}$ denotes the relative genus number, and $a_{K/F}$ denotes the ambiguous class number with respect to $K/F$, see e.g. this Yokoi's paper. My question is "What can we say about the finite abelian extensions?" Does still the equality $g_{K/F}=a_{K/F}$ hold in this case? If not, is there any explicit relation between $a_{K/F}$ and $g_{K/F}$? I guess, at least $a_{K/F} \mid g_{K/F}$ for $K/F$ a finite abelian extension of number fields, but I couldn't prove or disprove it.

Answer 1

If I understand the question correctly, the ambiguous classes are those fixed by $\text{Gal}(K/F)$, and the genus class number is the degree $(KF:K)$ of the maximal extension $KF/K$ that is unramified and for which $F/K$ is abelian.

In this case consider a number field $F$ with class group $[2,2]$ and let $K$ be the Hilbert class field of $F$. Assume moreover that $K$ has class number $2$, so the Hilbert class field $L$ of $K$ is dihedral or quaternion. Then $\text{Cl}(K)$ is ambiguous since the nontrivial ideal class is fixed by everything, so the ambiguous class number is $2$. On the other hand, $L$ is not the compositum of two abelian extensions $K$ and $F$, hence the genus class number is trivial.

For a similar example with $F = {\mathbb Q}$ consider $K = {\mathbb Q}(\sqrt{-3},\sqrt{13})$ with class number $2$. Again, the nontrivial ideal class is necessarily ambiguous, but the genus class number is $1$ since the Hilbert class field $L = K(\sqrt{-1+2\sqrt{-3}})$ is dihedral over ${\mathbb Q}$, and thus cannot be written as the compositum of $K$ and an abelian extension of ${\mathbb Q}$.