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 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 ≡ 2 (mod 3).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-3,
author = {Alain Kraus},
title = {Remarques sur le premier cas du th\'eor\`eme de Fermat sur les corps de nombres},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {133-141},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-3}
}
Alain Kraus. Remarques sur le premier cas du théorème de Fermat sur les corps de nombres. Acta Arithmetica, Tome 168 (2015) pp. 133-141. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-3/