On the Finiteness of Hyperelliptic Fields with Special Properties and Periodic Expansion of √f

Abstract

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.