# Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

## TL;DR

This paper extends the asymptotic Fermat's Last Theorem to many infinite families and numerous small degree number fields by integrating recent results with classical algebraic and Diophantine techniques.

## Contribution

It combines recent criteria with class field theory, p-group theory, and Diophantine approximation to prove the theorem for broad classes of number fields.

## Key findings

- Proves the asymptotic Fermat's Last Theorem for many infinite families of number fields.
- Establishes the theorem for thousands of small degree number fields.
- Provides effective results for fields like $ ext{Q}( ext{ extmu}_{2^r})^+$.

## Abstract

Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(\zeta_{2^r})^+$.