On the Finiteness of Hyperelliptic Fields with Special Properties and Periodic Expansion of √f
Vladimir Petrovich Platonov, Vladimir Sergeevich Zhgoon, М. М. Петрунин, Yu. N. Shteinikov · Doklady Mathematics · 2018
We prove the finiteness of the set of square-free polynomials f ∈ k[x] of odd degree distinct from 11 considered up to a natural equivalence relation for which the continued fraction expansion of the irrationality $$\sqrt {f\left( x \right)} $$ in k((x)) is periodic and the corresponding hyperelliptic field k(x)(√f) contains an S-unit of degree 11. Moreover, it was proved for k = ℚ that there are no polynomials of odd degree distinct from 9 and 11 satisfying the conditions mentioned above.