Fermat’s last theorem over ℚ(,)

Khawaja, Maleeha; Jarvis, Frazer

doi:10.2140/ant.2025.19.457

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Fermat's last theorem over $\mathbb{Q}(\sqrt{2}, \sqrt{3})$

Maleeha Khawaja and Frazer Jarvis

Vol. 19 (2025), No. 3, 457–480

DOI: 10.2140/ant.2025.19.457

Abstract

In this paper, we begin the study of the Fermat equation $x^{n} + y^{n} = z^{n}$ over real biquadratic fields. In particular, we prove that there are no nontrivial solutions to the Fermat equation over $ℚ (\sqrt{2}, \sqrt{3})$ for $n \geq 4$ .

Dedicated to Iffat (Zaman) Khawaja January 1936 – January 2022

Keywords

Fermat equation, modularity, Galois representations, rational points, elliptic curves

Mathematical Subject Classification

Primary: 11D41, 14G05, 14H52

Milestones

Received: 26 December 2022

Revised: 14 March 2024

Accepted: 29 April 2024

Published: 20 February 2025

Authors

Maleeha Khawaja
School of Mathematics and Statistics University of Sheffield Sheffield United Kingdom

Frazer Jarvis
School of Mathematics and Statistics University of Sheffield Sheffield United Kingdom

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 19, No. 3, 2025