Remarques sur le premier cas du th\'eor\`eme de Fermat sur les corps de nombres

TL;DR

This paper explores local obstructions to Fermat's Last Theorem over number fields, extending Sophie Germain's criteria to imaginary quadratic fields and providing conditions for primes congruent to 2 mod 3.

Contribution

It generalizes Sophie Germain's criterion to imaginary quadratic fields and offers a practical condition to verify the first case of Fermat's Last Theorem over number fields.

Findings

Established analogous Sophie Germain criteria for imaginary quadratic fields.

Provided a testable condition for primes p ≡ 2 mod 3 over number fields.

Extended classical results to broader algebraic settings.

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 has 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 ${\bf Q}$, we give an easily testable condition which allows occasionally to prove the first case of Fermat's Last Theorem over number fields for a prime number $p\equiv 2 \mod 3$.