TITLE: Number fields with only totally positive or totally negative units
QUESTION [6 upvotes]: Let $L$ be a totally real Galois extension, that is cyclic and of odd prime degree $p>2$. Let $\mathcal{O}_L$ be the ring of integers of $L$, and let $U=\mathcal{O}_L^\times$ be the unit group. The only roots of unity in such an extension are $\pm 1$, thus by Dirichlet's unit theorem, 
$$\mathcal{O}_L^\times \cong \langle -1 , u_1, u_2, u_3, \ldots, u_{p-1} \rangle$$
for some units $u_1,\ldots,u_{p-1}$ of infinite order.
Questions: 

Is it possible for all $u_1,\ldots, u_{p-1}$ to be totally positive units? (A unit $u$ is totally positive if all its embeddings $\tau(u)\in \mathbb{R}$ are positive.) 
If so, how often would one expect this to happen?
If it may occur, how would one detect this phenomenon without having to compute the entire unit group?

If $u$ is totally positive, then $-u$ is totally negative, so the question is asking whether there are such cyclic extensions $L/\mathbb{Q}$ with all units being totally positive or totally negative.

REPLY [4 votes]: A list of references concerning what has been done on this question may be found here; the question on how often this happens is being studied in this artcle. I guess you may assume that any questions not answered in these papers are open problems.