On Fermat's equation over quadratic imaginary number fields

Abstract

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