Primitive norm extraction and a signed descent for the cubic Fermat equation

We give a self-contained proof of the cubic case of Fermat's theorem through a signed form of Euler's descent, with the required extraction step proved directly in $\mathbb{Z}[\sqrt{-3}]$. A pigeonhole argument of Thue represents the relevant primes by $X^2+3Y^2$, and explicit division in $\mathbb{Z