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

The main result of the present article is a proof of Fermat’s Last Theorem for sufficiently large prime exponents $p$ with $p\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$ over certain number fields. 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.