Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 11G / c / Solution

Factor the equation in the Eisenstein integers:Let $\pi =1-\omega$. Its norm is $3$, andso $3$ is a unit times ${\pi }^2$. Suppose $x\equiv 2(\mathrm{mod}3)$, say $x=3k+2$. Since $\omega =1-\pi$,The first term is divisible by ${\pi }^2$ and the second has discrete valuation exactly one, so $v_{\pi }(x+\omega )=1$. Conjugation gives $v_{\pi }(x+{\omega }^2)=1$ as well. ConsequentlyBut $v_{\pi }(y^3)=3v_{\pi }(y)$ is divisible by three, a contradiction. Hence