Intrinsic construction of other Galois extensions

Question

Suppose L/F is a degree 4 Galois extension of global fields, with $Gal(L/F)=Z/2\times Z/2$. Then there are three degree 2 subextensions $K_i/F$ with $i=1,2,3$. Now start with $K_1/F$, how to give an intrinsic construction of the other two $K_i$'s? By intrinsic I mean not using the arithmetic of $L$.

Answer 1

Probably, such a construction does not exist. Consider the following case:

Let $F=\mathbb{Q}$, $p, q, r, s$ be four distinct positive rational primes, $K_1=\mathbb{Q}(\sqrt{pq})$, $K_2=\mathbb{Q}(\sqrt{rs})$, $K_3=\mathbb{Q}(\sqrt{pqrs})$, $L=\mathbb{Q}(\sqrt{pq}, \sqrt{rs})$.

Indeed $Gal(L/F)=\mathbb{Z}/2 \times \mathbb{Z}/2$, but given just, say, $K_1$, you cannot recover the others.

Answer 2

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.