Remarques sur le premier cas du théorème de Fermat sur les corps de nombres
Volume 167 / 2015
Acta Arithmetica 167 (2015), 133-141
MSC: Primary 11D41.
DOI: 10.4064/aa167-2-3
Abstract
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 $p\equiv 2\ {\rm mod}\ 3$.
