TITLE: Proving Fermat's Last Theorem for n=3 using Euler's and Lamé's approach?
QUESTION [9 upvotes]: 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?

REPLY [3 votes]: I recommend Harold Edwards' Fermat's Last Theorem. Euler's method of infinite descent for the case n = 3 (with a careful explanation of the gap in Euler's proof) is given and corrected in sections 2.2, 2.5 of this book. This also takes about 5 pages.  
This may be the version in Hardy and Wright but I doubt it. Edwards also explains that the idea can be corrected using an idea of Euler's, so the whole thing is essentially due to Euler.