Geodesic
Fermat’s Last Theorem over $\mathbb {Q}(\sqrt {\text{5}})$ and $\mathbb {Q}(\sqrt {\text{17}})$
Canadian journal of mathematics, Tome 76 (2024) no. 1, pp. 18-38

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DOI

We prove Fermat’s Last Theorem over $\mathbb {Q}(\sqrt {5})$ and $\mathbb {Q}(\sqrt {17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer–Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of $\mathbb {Q}(\sqrt {5})$ is a generalization to a real quadratic base field of the one used by Chen and Siksek. For the case of $\mathbb {Q}(\sqrt {17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett, Chen, Dahmen, and Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.
DOI : 10.4153/S0008414X22000633
Mots-clés : Fermat equation over quadratic fields, Galois representations, Hilbert modular forms 
Chen, Imin; Efemwonkieke, Aisosa; Sun, David. Fermat’s Last Theorem over $\mathbb {Q}(\sqrt {\text{5}})$ and $\mathbb {Q}(\sqrt {\text{17}})$. Canadian journal of mathematics, Tome 76 (2024) no. 1, pp. 18-38. doi: 10.4153/S0008414X22000633
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     title = {Fermat{\textquoteright}s {Last} {Theorem} over $\mathbb {Q}(\sqrt {\text{5}})$ and $\mathbb {Q}(\sqrt {\text{17}})$},
     journal = {Canadian journal of mathematics},
     pages = {18--38},
     year = {2024},
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