The question:

Factorise the polynomial $\displaystyle z^4 + 2z^2 - 3$ over the complex numbers.

My attempt:

Let $\displaystyle u = z^2$

$\displaystyle u^2 + 2u -3 = 0$

$\displaystyle \frac{-2 \pm \sqrt{4 - 4(1)(-3)}}{2}$

u = 1, -3

(u - 1)(u + 3) = 0

$\displaystyle (z^2 - 1)(z^2 + 3) = 0$

What do I do from here? Do I take z^2 = 1 (for example) then apply De'Movres theorem to find the roots of unity? Or is there a better way? Thanks.