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

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 \(K((x))\) .