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

 
PIIS086956520003431-7-1
DOI10.31857/S086956520003431-7
Publication type Article
Status Published
Authors
 
Affiliation: Scientific Research Institute for System Analysis, RAS
Address: Russian Federation
Affiliation: Scientific Research Institute for System Analysis, RAS
Address: Russian Federation
Affiliation: Scientific Research Institute for System Analysis, RAS
Address: Russian Federation
Affiliation: Scientific Research Institute for System Analysis, RAS
Address: Russian Federation
Journal nameDoklady Akademii nauk
EditionVolume 483 Issue 6
Pages609-613
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 √f(x) 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.

Keywords
Received26.12.2018
Publication date26.12.2018
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