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Fermat’s Last Theorem over
$\mathbb {Q}(\sqrt {\text{5}})$ and 
$\mathbb {Q}(\sqrt {\text{17}})$
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 21 November 2022, pp. 18-38
- Print publication:
- February 2024
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- Article
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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.
