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

TL;DR

This paper proves Fermat's Last Theorem for large primes congruent to 2 mod 3 over specific number fields, including maximal real subfields of certain cyclotomic extensions, by combining modular methods with generalized arithmetic results.

Contribution

It extends Fermat's Last Theorem to new classes of number fields using a novel combination of modular techniques and generalized arithmetic results.

Findings

Fermat's Last Theorem holds for large primes p ≡ 2 mod 3 over certain number fields.

The proof applies to maximal real subfields of cyclotomic extensions.

The approach generalizes previous arithmetic results to broader fields.

Abstract

The main result of the present article is a proof of Fermat's Last Theorem for sufficiently large prime exponents $p$ with $p \equiv 2 \pmod{3}$ 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.