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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2603.15065 |
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| _version_ | 1866914396357263360 |
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| author | Khawaja, Maleeha |
| author_facet | Khawaja, Maleeha |
| contents | Let $n\geq 3$ 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 $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15065 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Primitive points on some low degree Fermat curves Khawaja, Maleeha Number Theory Let $n\geq 3$ 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 $3$. |
| title | Primitive points on some low degree Fermat curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.15065 |