Proving Fermat's Last Theorem for n=3 using Euler's and Lamé's approach?

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 math.stanford.edu/~lekheng/flt ? –  lhf May 4 '12 at 13:30
According to Wikipedia, Euler used infinite descent, not $\mathbb Z[\omega]$. –  lhf May 4 '12 at 13:33

The proof takes 5 pages in Hardy and Wright, An Introduction to the Theory of Numbers (pages 248 to 253 in the 6th edition). No doubt it can be found in many other intro Number Theory texts, as well as on the web.

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Ribenboim's Fermat's Last Theorem for Amateurs is an excellent resource for this.

You can also check out this blog, but I always found it hard to navigate.

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