2024/04/13 by Alain Kraus, Kraus, Alain
Arts and Humanities · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2404.08922
openalex publication_date 2024/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ℚ be an algebraic closure of ℚ and ℚtr be the subfield of ℚ obtained by taking the union of all totally real number fields. For any prime p≥ 3, let Fp/ℚ be the Fermat curve of equation xp+yp+zp=0. In 1996, Pop has shown that the field ℚtr is large. In particular, the set Fp(ℚtr) of the points of Fp rational over ℚtr is infinite. How to explicit non-trivial points (xyz≠ 0) in Fp(ℚtr) ? If one has p≥ 5, it seems that the only points already known in Fp(ℚtr) are those of Fp(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 ℚ is at most 5 over which F5 has non-trivial points. We propose here to explicit infinitely many points of F5 rational over totally real fields of degree 6 over ℚ.