Hi folks,

there is a very well worn method of finding the cube roots of unity, but there are always more than one way of doing things, so I tried a few, hoping that all roads lead to the mathematical rome.

$z^3 = 1$

$(x + iy)^3 = 1$

$x^3 - 3xy^{2} + 3x^{2}yi -iy^{3} = 1$

$(x^3 - 3xy{^2}) + i(3x{^2}y - y^3) = 1$

so

$x^3 - 3xy^{2} = 1$ ...... (1)

$3x^{2}y - y^3 = 0$ .......(2)

from (2)

$y ( y^2 - 3x^2) = 0$

$ y^2 = 3x^2$

from (1)

$x ( x^2 - 3y^2) = 1$

$x = 1$ or $x^2 = 3y^2 + 1$

$x^2 = 9x^2 + 1$

$-8x^2 = 1$

this is not going well, but I can't see the flaw in it. Where am I going wrong?