Sur le th\'eor\`eme de Fermat sur ${\bf Q}(\sqrt{5})$

TL;DR

This paper presents a modular approach to verify Fermat's Last Theorem over the quadratic field Q(√5) for large primes, using Wendt's resultant and providing practical criteria for specific prime ranges.

Contribution

Introduces an easy, testable modular criterion based on Wendt's resultant to prove Fermat's Last Theorem over Q(√5) for primes less than 10^7.

Findings

Proves Fermat's Last Theorem over Q(√5) for primes p<10^7.

Develops a criterion related to Wendt's resultant for testing FLT over quadratic fields.

Provides analogous results for Sophie Germain type conditions.

Abstract

Let $p$ be an odd prime number. Using modular arguments, we give an easy testable condition which allows often to prove Fermat's Last Theorem over the quadratic field ${\bf Q}(\sqrt{5})$ for the exponent $p$. It is related to the Wendt's resultant of the polynomials $X^n-1$ and $(X+1)^n-1$. We deduce Fermat's Last Theorem over this field in case one has $5\leq p<10^7$, and we obtain analogous results on Sophie Germain type criteria.