I am posting this question on MO since I haven't received any answers on MSE.

Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about elliptic curves.

Start by noticing $x^3=(1-y)(1+y+y^2)$ in $\Bbb C[x,y]/(x^3+y^3-1)$. I am trying to show first that $x$ is irreducible modulo $(x^3+y^3-1)$.

Every polynomial in $\Bbb C[x,y]$ is of the form $$p_1(x)+p_2(x)y+p_3(x)y^2+r(x,y)(x^3+y^3-1)$$ for some polynomials $p_1(x),p_2(x),p_3(x),r(x,y)$ in $\Bbb C[x,y]$. So that given $$x=(p_1(x)+p_2(x)y+p_3(x)y^2)(q_1(x)+q_2(x)y+q_3(x)y^2)+r(x,y)(x^3+y^3-1)$$ one needs to show that either $q_1(x)+q_2(x)y+q_3(x)y^2$ or $p_1(x)+p_2(x)y+p_3(x)y^2$ is a unit modulo $(x^3+y^3-1)$, i.e for example $$(p_1(x)+p_2(x)y+p_3(x)y^2)f(x,y) = k+g(x,y)(x^3+y^3-1)$$ for some constant $k$ and polynomials $f,g\in\Bbb C[x,y]$ is the same as saying $p_1(x)+p_2(x)y+p_3(x)y^2$ is a unit modulo $(x^3+y^3-1)$.

This gets extremely messy afterwards. Is there a better way I could approach this?