On fermat’s last theorem for n = 3 and n = 4

A solution to Fermat’s equation, xn + yn = zn, is called trivial if xyz = 0. In this paper we will prove Fermat’s Last Theorem, which states all rational solutions are trivial for n > 2, when 3 | n or 4 | n. For n = 3 we will show all solutions in the Eisenstein Field, Q( √ −3), are trivial. Our proof is in the same vain as Gauss’ proof, but argued towards a different contradiction. For n = 4 we will show all solutions in the Gaussian Field, Q(i), are trivial. We will follow Hilbert’s proof given in [3] which has the flavor of argument made by Gauss with the Eisenstein Field.