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.
Categories: math.NT