On the Sequences of Polynomials $$\boldsymbol{f}$$ with a Periodic Continued Fraction Expansion $$\sqrt{\boldsymbol{f}}$$

Abstract

For each \(n\geqslant 3\), 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 \(n\geqslant 5\), 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 \(K((x))\).