Abstract

We obtain a complete description of cubic polynomials f over algebraic number fields \(\mathbb{K}\) of degree \(3\) over \(\mathbb{Q}\) for which the continued fraction expansion of \(\sqrt f \) in the field of formal power series \(\mathbb{K}((x))\) is periodic. We also prove a finiteness theorem for cubic polynomials \(f \in K[x]\) with a periodic expansion of \(\sqrt f \) for extensions of \(\mathbb{Q}\) of degree at most 6. Additionally, we give a complete description of such polynomials f over an arbitrary field corresponding to elliptic fields with a torsion point of order \(N \geqslant 30\).

中文翻译：

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代数数域上三次多项式的展开次数为$$ \ sqrt f $$的连续分数的有限性

摘要

我们获得三次多项式的完整描述˚F超过代数数字段\（\ mathbb {K} \）程度的\（3 \）超过\（\ mathbb {Q} \）对于其中的连分数膨胀\（\ SQRT f \）在形式幂级数\（\ mathbb {K}（（x））\）的领域中是周期性的。我们还证明了三次多项式\ [f \ in K [x] \]的有限性定理，其中\（\ mathbb {Q} \）的度数扩展最多为6的周期扩展为\（\ sqrt f \）。此外，我们给出了在具有扭转阶数的椭圆形域对应的任意域上此类多项式f的完整描述\（N \ geqslant 30 \）。