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Euler and Lamé are said to have proven FLT for $n=3$ that is, they are believed to have shown that $x^3 + y^3 = z^3$ has no nonzero integer solutions. According to Kleiner they approached this by decomposing $x^3 + y^3$ into $(x + y)(x + y\omega)(x + y\omega^2)$ where $\omega$ is the primitive cube root of unity or $w = \frac{-1 + \sqrt{3}i}{2}$.

How would you finish the rest of the proof?

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    By Kleiner, do you mean Israel Kleiner, "From Fermat to Wiles: Fermat's Last Theorem becomes a theorem," Elemente der Mathematik, 55 (1), February 2000, pp. 19–37 , available at http://math.stanford.edu/~lekheng/flt/ ?2012-05-04
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    According to [Wikipedia](http://en.wikipedia.org/wiki/Proof_of_Fermat's_Last_Theorem_for_specific_exponents#n.C2.A0.3D.C2.A03), Euler used infinite descent, not $\mathbb Z[\omega]$.2012-05-04

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