On Fermat's Last Theorem over the ${\mathbb{Z}}_3$-extension of $\mathbb{Q}$ and other fields

Abstract

The main result of the present article is a proof that, over certain totally real number fields $K$, the Fermat equation $x^p+y^p+z^p=0$ has no non-trivial primitive solutions in ${\mathcal{O}}_K$ for sufficiently large prime exponents $p$ with $p\equiv 2(\mathrm{mod}3)$. A particular case of these fields are the maximal real subfields of the cyclotomic extensions $\mathbb{Q}({\zeta }_{3^n})$ for every $n$. Our strategy consists in combining the modular method with a generalization of an arithmetic result of Pomey to these fields.

arXiv comment14 pages