Remarques sur le premier cas du théorème de Fermat sur les corps de nombres

Volume 167 / 2015

Alain Kraus Acta Arithmetica 167 (2015), 133-141 MSC: Primary 11D41. DOI: 10.4064/aa167-2-3


The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain obtained an important result in this direction by establishing that, if $2p+1$ is a prime number, the first case of Fermat's Last Theorem is true for $p$. In this paper, we investigate such obstructions over number fields. We obtain analogous results on Sophie Germain type criteria, for imaginary quadratic fields. Furthermore, extending a well known statement over ${{\mathbb Q}}$, we give an easily testable condition which allows one occasionally to prove the first case of Fermat's Last Theorem over number fields for a prime number $p\equiv 2\ {\rm mod}\ 3$.


  • Alain KrausÉquipe de Théorie des Nombres
    Institut de Mathématiques de Jussieu
    Université de Paris VI
    4 Place Jussieu
    75005 Paris, France

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