By considering z[sqrt[-2]] show that x^2+2=y^3 only has two integer solutions, (+/-5,3)
I can see that N(x+i Sqrt[2])=y^3, I think x+i Sqrt[2] is prime in z[i Sqrt[2]] so y^(3/2) must be x+i Sqrt[2] but I don't really see how any of this helps. I think I'm missing something.