Abstract:
The paper [1] raised the question of the finiteness of the number
squarefree polynomials $f \in \mathbb{Q}[h]$ of fixed degree
with periodic continued fraction expansion of $\sqrt{f\mathstrut}$ in the field $\mathbb{Q}((h))$,
for which the fields $\mathbb{Q}(h)(\sqrt{f\mathstrut})$ are not isomorphic to
one another and to fields of the form $\mathbb{Q}(h)(\sqrt{ch^{n} + 1})$,
where $c \in \mathbb{Q}^{\ast}$, $n \in \mathbb{N}$.
In the joint paper, V.P. Platonov and G.V. Fedorov [2]
obtained a positive answer to this question for elliptic fields
$\mathbb{Q}(h)(\sqrt{f\mathstrut})$, $\deg f = 3$. The report will present the results of
the article [2].
[1] V.P. Platonov, G.V. Fedorov, On the periodicity of continued fractions in elliptic fields.
Doklady Mathematics. 2017 (to appear).
[2] V.P. Platonov, G.V. Fedorov, On the periodicity of continued fractions in hyperelliptic fields.
Doklady Mathematics. 2017 (to appear).
Contact us: