  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 $p \geq 5$, Fermat's equation
$a^p+b^p+c^p=0$ does not have non-trivial solutions over $\mathbb Q(\sqrt{-2})$
and $\mathbb Q(\sqrt{-7})$.
