Given some $e_i=0$ or $1$ for $1\le i \le 3$.

I wish to construct a totally real cubic number field $K$ so that $\prod_{i=1}^3 sgn(\sigma_i(u))^{e_i}$ is always $1$ for any $u\in O_K^\times$ where $\sigma_i$'s are elements in the Galois group.

For quadratic case this question is somehow equivalent to Pell's equation. I am interested in higher degree cases.

Say $e_1=0$, $e_2=e_3=1$. Of course, one can construct a field $K$ such that all of the elements in $O_K^\times$ is either totally positive or totally negative. I am interested in whether we can construct $K$ with $\sigma_1(u)>0, \sigma_2(u)\sigma_3(u)>0$ for all $u\in O_K^\times$ up to multiplication by $-1$, but there exists some $u$ with $\sigma_1(u)>0, \sigma_2(u)<0, \sigma_3(u)<0$.

Sorry for being wordy. Thank you in advance!

Given some $e_i=0$ or $1$ for $1\le i \le 3$.

I wish to construct a totally real cubic number field $K$ so that $\prod_{i=1}^3 sgn(\sigma_i(u))^{e_i}$ is always $1$ for any $u\in O_K^\times$ where $\sigma_i$'s are real embeddings of $K$.

For quadratic case this question is somehow equivalent to Pell's equation. I am interested in higher degree cases.

Say $e_1=0$, $e_2=e_3=1$. One may want to construct a field $K$ such that all of the elements in $O_K^\times$ is either totally positive or totally negative. I would like to see such an example.

Furthermore, I am interested in whether we can construct $K$ with $\sigma_1(u)>0, \sigma_2(u)\sigma_3(u)>0$ for all $u\in O_K^\times$ up to multiplication by $-1$, but there exists some $u$ with $\sigma_1(u)>0, \sigma_2(u)<0, \sigma_3(u)<0$.

Sorry for being wordy. Thank you in advance!