Let $K$ be a primitive (i.e. not biquadratic) quartic CM-field. That is we have $[K:\mathbb{Q}]=4$ and let $K_0=\mathbb{Q}(\sqrt{d})$ be the totally real quadratic subfield, here $d> 1$, that is we have $[K:K_0]=[K_0:\mathbb{Q}]=2$.

Denote by $O_K^{\times}$ the set of units in $K$ and by $(O_{K_0}^{\times})^{+}$ the set of totally positive units in $K_0$. Then $N_{K/K_0}(O_K^{\times})$ is a subgroup of $(O_{K_0}^{\times})^{+}$

$\textbf{Question:}$ Can we classify all such fields $K$ with the property $r=[(O_{K_0}^{\times})^{+}:N_{K/K_0}(O_K^{\times})]=1$?

a) If $\epsilon_0$ denotes the fundamental unit of $K_0$ and we have $N_{K_0/\mathbb{Q}}(\epsilon_0)=-1$ then $r=1$. So these are all $K$ with the property that $K_0$ has a (fundamental) unit of norm -1. Are these exactly the fields $\mathbb{Q}(\sqrt{t^2+4})$ for some $t\in \mathbb{Z}$?

b)What happens if $N_{K_0/\mathbb{Q}}(\epsilon_0)=1$? Is it possible that there is a unit $\epsilon\in K$ such that $\epsilon_0=N_{K/K_0}(\epsilon)$? Then we also should get $r=1$. So are the there such fields with a unit $\epsilon$ such that $N_{K/K_0}(\epsilon)$ is a fundamental unit of $K_0$ with norm 1? Can we classify all such $K$?

c) Are there any other $K$ with $r=1$?