Primitive points on some low degree Fermat curves

Abstract

Let $n\ge 3$ be an integer. Let $F_n$ be the Fermat curve defined by the Fermat equation $x^n+y^n=z^n$. For a curve $C/\mathbb{Q}$, we say an algebraic point $P\in C(\bar{\mathbb{Q}})$ is primitive if the Galois group of the Galois closure of the number field $\mathbb{Q}(P)$ is a primitive permutation group. Recall that $A_4$ is a primitive subgroup of $S_4$. We prove that there are no non-trivial quartic points on $F_n$ with Galois closure $A_4$, when $n=7$ and $n=8$. We also provide sufficient conditions for the non-existence of non-trivial points on the Fermat curves $F_6$ and $F_8$ defined over a given primitive number field of degree at least $3$.