Let $p$ be an odd prime and $\zeta$ be a primitive $p$-th root of unity. Let $K^{+}=\mathbb{Q}(\zeta+\zeta^{-1})$ denote the maximal real subfield of $K=\mathbb{Q}(\zeta)$. For $2 \leq j \leq (p-1)/2$, let $\eta_j=\frac{\zeta^{j/2}-\zeta^{-j/2}}{\zeta^{1/2}-\zeta^{-1/2}}$ and let $C$ denote the group generated by the $\eta_j$ (the real cyclotomic units modulo $\{\pm 1\}$) and let $\{\varepsilon_j\}_{j=1}^{(p-3)/2}$ denote a system of fundamental system of units for $K^{+}$ and denote the group generated by the $\varepsilon_j$ by $E$ (the real units modulo $\{\pm 1\}$). It is well known that $\vert E/C \vert = h$, the class number of $K^{+}$.
$h$ is notoriously difficult to compute and it seems (but I may not be up-to-date) that there are no known examples of $K^{+}$ for which $h>1$, although it has been proved for a few primes ($163$,$191$, and $229$) that $h>1$ (along with the predicted values) assuming the Generalized Riemann Hypothesis. I read recently elsewhere that there are no known examples of units which are not cyclotomic units. If it has not been proven for any $K^{+}$ that $h>1$, then this is necessarily the case, but would this still be the case if we did know that $h>1$ for some $K^{+}$? For example, if we knew that $h=4$ for $p=163$ (this is the predicted $h$ given the Generalized Riemann Hypothesis), could we use this to explicitly produce a unit not lying in $C$ without radicals? I say without radicals because I believe given $h$, we could find some unit of the form $(\prod_{j}\eta_j^{a_j})^{1/h}$ (with not all $a_j \equiv 0 (\text{mod} \ h)$) lying in $E$ but not $C$.