Let $\zeta_{12}$ be the $12$-th root of unity. Consider the field $\mathbb{Q}(\zeta_{12})$ and the subfield $\mathbb{Q}(\sqrt{3})$.
Let $u$ be a fundamental unit of $\mathbb{Q}(\zeta_{12})$, then $u\overline{u}=|u|^{2}\in\mathbb{R}$. Are we able to conclude that $u\overline{u}$ is a unit in $\mathbb{Q}(\sqrt{3})$?
I was going through some papers and confronted this statement in the proof that $\zeta_{12}-1$ is a fundamental unit in $\mathbb{Q}(\zeta_{12})$. Out of curiousity I tried to figure out why this is true, but can't think of a way how to prove this. (Example 4.11 of the following paper:)
https://kconrad.math.uconn.edu/blurbs/gradnumthy/unittheorem.pdf
Any help would be appreciated!