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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.

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    $\begingroup$ This can happen for $p=2$, so I don't see a reason a priori that this shouldn't happen for $p>2$. I estimate that if the signs of the embeddings of each unit are uniformly distributed (fix the first embedding to be positive), all generators should be totally positive with probability $(1/16)^4$? $\endgroup$ Jul 26, 2017 at 17:49
  • $\begingroup$ Four fundamental units? Not $p-1$? $\endgroup$ Jul 26, 2017 at 21:35
  • $\begingroup$ This will happen if and only if $H^{\mathrm{nar}}/H$ has degree $2^{p-1}$, where $H$ is the Hilbert class field of $L$ and $H^{\mathrm{nar}}$ is the narrow Hilbert class field. Don't know if that helps answer any of your questions though. $\endgroup$ Jul 26, 2017 at 21:37
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    $\begingroup$ There is an exact sequence $1 \rightarrow E/E^{\mathrm{tp}} \rightarrow \oplus_{v \in \mathrm{Pl}_{\infty}^{r}} \{ \, \pm 1 \, \} \rightarrow \mathrm{Gal}(H^{\mathrm{nar}}/H) \rightarrow 1$. Here, $E^{\mathrm{tp}}$ is the group of totally positive units in $L$, the direct sum is over the real places of $L$, and the last arrow is the global reciprocity map from the idelic formulation of class field theory. See, for instance, Chapter III Theorem 1.1 of Gras "Class Field Theory" and also the remark 1.1.5 following it. $\endgroup$ Jul 27, 2017 at 0:45
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    $\begingroup$ THANK YOU! This is very helpful. $\endgroup$ Jul 27, 2017 at 1:48

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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.

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