Let p be an odd natural number ≥3. Inspired by results from Euclid's Elements, we express the irrational$y=\sqrt[p]{d+\sqrt{R}},$ whose degree is 2p, as a polynomial function of irrationals of degrees ≤p. In certain cases y is expressed by simple radicals. This reduction of the degree exhibits remarkably regular patterns of the polynomials involved. The proof is based on hypergeometric summation, in particular, on Zeilberger's algorithm.

中文翻译：

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以欧几里得精神还原自由基

令p为≥3的奇数自然数。受欧几里得元素研究结果的启发，我们表达了非理性$ÿ=\sqrt[p]{d+\sqrt{\mathrm{[R}}}，$其度为2 p，作为度≤的无理的多项式函数p。在某些情况下，y由简单的部首表示。这种程度的降低表现出所涉及的多项式的明显规则的模式。该证明基于超几何求和，特别是基于Zeilberger算法。