曲线$x^5+y^5+z^5=0$的全实点
Points totalement réels de la courbe $x^5+y^5+z^5=0$
AI总结：
本文针对5次费马曲线$F_5$的全实非平凡点构造问题，证明不存在次数≤5的全实数域使$F_5$有非平凡全实点，并给出无穷多个6次全实数域上的$F_5$有理点的显式构造，推进了全实域上费马曲线非平凡点的研究。
摘要：
设$\overline{\mathbb{Q}}$为$\mathbb{Q}$的代数闭包，$\mathbb{Q}^{tr}$为所有全实数域的并构成的$\overline{\mathbb{Q}}$的子域。对任意素数$p\geq 3$，记$F_p/\mathbb{Q}$为方程$x^p+y^+z^p=0$对应的费马曲线。1996年，Pop证明了域$\mathbb{Q}^{tr}$是大域，特别地，$F_p$在$\mathbb{Q}^{tr}$上的有理点集$F_p(\mathbb{Q}^{tr})$是无限集。如何显式给出$F_p(\mathbb{Q}^{tr})$中的非平凡点（满足$xyz\neq 0$）？当$p\geq 5$时，目前已知的$F_p(\mathbb{Q}^{tr})$中的点似乎仅有$F_p(\mathbb{Q})$中的点，且均为平凡点。本文研究$p=5$情形下的该问题：不存在$\mathbb{Q}$上次数不超过5的全实数域使得$F_5$在其上有非平凡点。本文提出显式构造无穷多个$F_5$在$\mathbb{Q}$上6次全实数域上的有理点的方法。
英文摘要：
Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ and $\mathbb{Q}^{tr}$ be the subfield of $\overline{\mathbb{Q}}$ obtained by taking the union of all totally real number fields. For any prime $p\geq 3$, let $F_p/\mathbb{Q}$ be the Fermat curve of equation $x^p+y^p+z^p=0$. In 1996, Pop has shown that the field $\mathbb{Q}^{tr}$ is large. In particular, the set $F_p(\mathbb{Q}^{tr})$ of the points of $F_p$ rational over $\mathbb{Q}^{tr}$ is infinite. How to explicit non-trivial points $(xyz\neq 0$) in $F_p(\mathbb{Q}^{tr})$ ? If one has $p\geq 5$, it seems that the only points already known in $F_p(\mathbb{Q}^{tr})$ are those of $F_p(\mathbb{Q})$ and they are trivial. In this paper, we investigate this question in case $p=5$. There are no totally real fields whose degree over $\mathbb{Q}$ is at most $5$ over which $F_5$ has non-trivial points. We propose here to explicit infinitely many points of $F_5$ rational over totally real fields of degree $6$ over $\mathbb{Q}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
