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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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.

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.

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.