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 Q ( i ) . Under the same assumption, we also prove that, for all prime exponents p ≥ 5 , Fermat's equation a p + b p + c p = 0 does not have non-trivial solutions over Q ( - 2 ) and Q ( - 7 ) .