  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.
