Fermat's Last Theorem over ${\mathbb Q}(\sqrt{5})$ and ${\mathbb Q}(\sqrt{17})$

TL;DR

This paper proves Fermat's Last Theorem over specific real quadratic fields, ${ m Q}(\sqrt{5})$ and ${ m Q}(\sqrt{17})$, for prime exponents in certain congruence classes, using modular methods and explicit reciprocity constraints.

Contribution

It extends the modular approach and reciprocity constraints to real quadratic fields, generalizing previous methods to new base fields.

Findings

Fermat's Last Theorem holds over ${ m Q}(\sqrt{5})$ and ${ m Q}(\sqrt{17})$ for certain primes.

Introduces explicit reciprocity constraints using quadratic reciprocity and Hilbert symbols.

Generalizes previous reciprocity methods to real quadratic fields.

Abstract

We prove Fermat's Last Theorem over ${\mathbb Q}(\sqrt{5})$ and ${\mathbb Q}(\sqrt{17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer-Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of ${\mathbb Q}(\sqrt{5})$ is a generalization to a real quadratic base field of the one used by Chen-Siksek. For the case of ${\mathbb Q}(\sqrt{17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett-Chen-Dahmen-Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.