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

Alain Kraus¹

¹ Équipe de Théorie des Nombres Institut de Mathématiques de Jussieu Université de Paris VI 4 Place Jussieu 75005 Paris, France

Acta Arithmetica, Tome 167 (2015) no. 2, pp. 133-141.

Résumé

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 $p\equiv 2\ {\rm mod}\ 3$.

Détail

DOI : 10.4064/aa167-2-3

Mots-clés : first fermats theorem prime exponent sometimes proved using existence local obstructions sophie germain obtained important result direction establishing prime number first fermats theorem paper investigate obstructions number fields obtain analogous results sophie germain type criteria imaginary quadratic fields furthermore extending known statement mathbb easily testable condition which allows occasionally prove first fermats theorem number fields prime number equiv mod