On the Finiteness of Hyperelliptic Fields with Special Properties and Periodic Expansion of $\sqrt f$

We prove the finiteness of the set of square-free polynomials $f \in k[x]$ of odd degree distinct from 11 considered up to a natural equivalence relation for which the continued fraction expansion of the irrationality $\sqrt{f(x)}$ in $k((x))$ is periodic and the corresponding hyperelliptic field $k(x)(\sqrt f)$ contains an $S$-unit of degree 11. Moreover, it was proved for $k = \mathbb{Q}$ that there are no polynomials of odd degree distinct from 9 and 11 satisfying the conditions mentioned above.