Abstract

This paper contains an explanation of Ramanujan-type formulas with cubic radicals of cubic irrationalities in the situation when these radicals are contained in a pure cubic extension. We give a complete description of formulas of such type, answering the Zippel’s question. It turns out that Ramanujan-type formulas are in some sense unique in this situation. In particular, there must be no more than three summands in the right-hand side and the norm of the irrationality in question must be a cube. In this situation we associate cubic irrationalities with a cyclic cubic polynomial, which is reducible if and only if one can simplify the corresponding cubic radical. This correspondence is inverse to the so-called Ramanujan correspondence defined in the preceding papers, where one associates a pure cubic extension to some cyclic polynomial.

中文翻译：

---

立方自由基的Ramanujan去嵌套公式

摘要

本文包含对Ramanujan型公式的解释，这些公式具有立方非理性的立方基团，当这些基团包含在纯立方扩展中时。我们回答了Zippel的问题，给出了此类公式的完整描述。事实证明，在这种情况下，Ramanujan型公式在某种意义上是唯一的。特别是，在右侧不得超过三个求和项，并且所讨论的非理性性的规范必须是一个立方体。在这种情况下，我们将三次非理性与循环三次多项式相关联，当且仅当一个人可以简化相应的三次自由基，才能将其简化。这种对应关系与先前论文中定义的所谓拉曼努扬对应关系相反，在Ramanujan对应关系中，将纯三次扩展与某个循环多项式相关联。