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arXiv · 1407.4435

Fermat's Last Theorem over some small real quadratic fields

Nuno Freitas·Samir Siksek
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Abstract

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the Fermat equation $x^n+y^n=z^n$ has no non-trivial solutions over the quadratic field $\mathbb{Q}(\sqrt{d})$ for $n \ge 4$. Furthermore, we show for $d=17$ that the same holds for prime exponents $n \equiv 3$, $5 \pmod{8}$.

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Nuno Freitas, Samir Siksek. 2014-09-18. Fermat's Last Theorem over some small real quadratic fields. https://doi.org/10.2140/ant.2015.9.875

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