On the sequences of polynomials $f$ with a periodic continued fraction expansion $\sqrt{f}$

G.V. Fedorov

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2024), pp. 25-30

Résumé

For each $n \ge 3$, three nonequivalent polynomials $f \in \mathbb{Q}[x]$ of degree $n$ were previously constructed for which $\sqrt{f}$ has a periodic continued fraction expansion in the field $\mathbb{Q}((x))$. In this paper, for each $n \ge 5$, two new polynomials $f \in K[x]$ of degree $n$ are found, defined over the field $K$, $[K : \mathbb{Q}] = [(n-1)/2]$, for which $\sqrt{f}$ has a periodic continued fraction expansion in the field $K((x))$.