# Quartic points on the Fermat quintic

## TL;DR

This paper characterizes algebraic points of degree 4 on the Fermat quintic curve, providing a detailed algebraic description and identifying the unique Galois extension associated with such points.

## Contribution

It offers a new algebraic description of degree 4 points on the Fermat quintic and establishes the uniqueness of the Galois extension involved.

## Key findings

- Only one Galois extension of degree 4 arises from these points.
- Provides a complete algebraic description of degree 4 points.
- Builds on previous geometric and diophantine work to classify points.

## Abstract

In this paper, we study the algebraic points of degree $4$ over $\mathbb{Q}$ on the Fermat curve $F_5/\mathbb{Q}$ of equation $x^5+y^5+z^5=0$. A geometrical description of these points has been given in 1997 by Klassen and Tzermias. Using their result, as well as Bruin's work about diophantine equations of signature $(5,5,2)$, we give here an algebraic description of these points. In particular, we prove there is only one Galois extension of $\mathbb{Q}$ of degree $4$ that arises as the field of definition of a non-trivial point of $F_5$.