Abstract

We obtain a complete description of fields \(\mathbb{K}\) that are quadratic extensions of \(\mathbb{Q}\) and of cubic polynomials \(f \in \mathbb{K}[x]\) for which a 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 \(f \in \mathbb{K}[x]\) with a periodic expansion of \(\sqrt f \) over cubic and quartic extensions of \(\mathbb{Q}\).

中文翻译：

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关于数域上三次多项式$$ \ sqrt f $$的连续分数展开的周期性问题

摘要

我们获得字段的完整说明\（\ mathbb {K} \）是的二次延伸\（\ mathbb {Q} \）和三次多项式的\（F \在\ mathbb {K} [X] \）为在正规幂级数\（\ mathbb {K}（（x））\）领域中\（\ sqrt f \）的连续分数扩张是周期性的。我们还证明了三次多项式\（f \ in \ mathbb {K} [x] \）的有限性定理，其中\（\ sqrt f \）在\（\ mathbb {Q} \ ）。