Rank of the group generated by the images of a unit in the ring of integers of totally real number fields

Question

Let $F$ be a totally real number field over $\mathbb Q$ of degree $m$ and let $\mathcal O_F$ be its ring of integers. By Dirichlet's unit theorem we know that $\mathcal O_F^* \cong \{\pm 1\} \times \mathbb Z^{m-1}$. Let $u \in \mathcal O_F^*$ be of infinite order. I'm interested in the group generated by $\{ \sigma_j(u) \mid 1\le j \le m \}$ where $\sigma_j$ are the $m$ real embeddings of $F$. Is this group always of rank $m-1$ (after dividing out potential torsion)?

Answer 1

The answer is no. Take a totally real quartic field with a quadratic subfield, and let $u$ be the fundamental unit of the quadratic subfield. For fields of prime degree, google for Minkowski units and check out Narkiewicz's book on algebraic number theory.