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?



Dihedral fields $L/\mathbb{Q}$ of degree $2^{k+1}$
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. 


