On the sequences of polynomials \(\boldsymbol{f}\) with a periodic continued fraction expansion \(\sqrt{\boldsymbol{f}} \) (Q6600277)

Using the characterization established in [\textit{V. P. Platonov} and the author, Sb. Math. 209, No. 4, 519--559 (2018; Zbl 1445.11135); translation from Mat. Sb. 209, No. 4, 54--94 (2018)], the author exhibits for each \(n \geq 5\) and via an algorithmic constructive method, new square-free polynomials \(f\in K[x]\) of degree \(n\) for which \(\sqrt{f}\) has a periodic continued fraction expansion in the field \(K((x))\), where \(K\) is a number field of degree \([K : \mathbb{Q}] = \lfloor (n- 1)/2 \rfloor \).\N\NIn particular, the polynomials found in this paper completes the ones exhibited in the previous cited paper for which \(K=\mathbb{Q}\); moreover, explicit examples are given for \(n=5\) and \(n=6\).