nt.number theory - Intrinsic construction of other Galois extensions

Let me first give a formulation of the question that makes a little bit of sense.
Let $K/F$ be a quadratic extension of a number field, and let $L/K$ be a class field
for an ideal group $D_L$ with index $2$ in the group of all idealc coprime to some
defining modulus.

1. Which properties of $D_L$ guarantee that $L/F$ is biquadratic?


2. If $L/F$ is biquadratic, how can we realize the intermediate fields
   $K_1$ and $K_2$ different from $K$ as class fields over $F$?

For base field $F = {mathbb Q}$, these questions are answered (in the classical
ideal language) in Cohn's A classical invitation to algebraic numbers and
class fields, Springer-Verlag 1978: see in particular Thm. 18.17, where
the necessary and sufficient condition for 1. to hold (assuming that $L/F$ is
normal, which happens if and only if the ideal group $D_L$ is invariant under
Gal$(K/F)$; see Cohn, Thm. 18.13) is that the ideal group contain a ring class
group modulo some conductor $f$, from which the fields $K_1$ and $K_2$ can then
be determined.

More generally, given an class field $K_1/F$ for the ideal group $D_1$, you can
use the translation theorem of class field theory to realize the compositum
$L = K_1K$ as a class field over $K$ by taking the group of ideals whose norms
land in $D_1$. This result is contained in most classical sources on class field
theory, quite likely also in Janusz.

Since there is a change of conductors involved when pushing an extension up some
tower, it is perhaps better to use the language of idèles, but I'm more
fluent in ideals.

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