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Let $L/\mathbb{Q}$ be a dihedral extension of degree $2^{k+1}$, namely that $L$ is Galois over $\mathbb{Q}$ and $\text{Gal}(L/\mathbb{Q}) \cong D_{2^k}$, the dihedral group on $2^{k+1}$ elements.

There is a unique quadratic subfield $K$ of $L$ such that $L/K$ is a cyclic degree $2^k$ extension when $k \geq 2$, corresponding to the unique index-2 cyclic subgroup.

My questions are:

1) For a fixed real quadratic field $K$, how do we parametrize all such $L$ satisfying the above properties, for fixed $k$?

2) How do we determine when $L/K$ is an unramified extension of number fields?

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  • $\begingroup$ The statement "There is a unique quadratic subfield such that $L/K$ is cyclic" is true for $k+1\ge 3$ but false for $k+1=2$, in which case there are 3 such subfields. Indeed, the number of cyclic subgroups of index 2 in $D_{2^k}$ (dihedral of order $2^{k+1}$) is $1$ for $k+1\ge 3$ (or $k+1=1$) but is $3$ for $k+1=2$. $\endgroup$ – YCor Oct 11 '19 at 14:24
  • $\begingroup$ You are right; when $k + 1 = 2$ the group is the Klein 4-group which is abelian and contains three quadratic subfields. I meant to exclude this case. I will edit the question to reflect this. $\endgroup$ – Stanley Yao Xiao Oct 11 '19 at 15:42
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1) I don't know whether there is a parametrization. For number theorists, such parametrizations are of little help since, while it is easy to find examples of such extensions, it is next to impossible to check whether a given extension belongs to the parametrized family, and if yes, for which choice of the parameters,

2) We compute the relative discriminant and check whether it is trivial. But I guess that wasn*t what you meant to ask. Reichardt [Zur Struktur der absoluten Idealklassengruppe im quadratischen Zahlkörper; J. reine Angew. Math. 170, 75-82 (1933)] showed how to construct the class fields you are looking for step by step. At each step of the tower you have to solve an equation of the form $ax^2 - by^2 = z^2$ with coefficients lying in the last field you have constructed.

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