On Fermat's equation over some quadratic imaginary number fields

Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat’s Last Theorem over $\mathbb{Q}(i)$. Under the same assumption, we also prove that, for all prime exponents $p\ge 5$, Fermat’s equation $a^p+b^p+c^p=0$ does not have non-trivial solutions over $\mathbb{Q}(\sqrt{-2})$ and $\mathbb{Q}(\sqrt{-7})$.