Fermat's Last Theorem over $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{17})$

Abstract

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-Siksek. For the case of $\mathbb{Q}(\sqrt{17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett-Chen-Dahmen-Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.