Let $\displaystyle
w = \frac {-1}{2} + \frac {3^{1/2}}{2}i
$, find units of Z[w].
1 is the unity, so it is a unit. And what else...?
The the previous excercises $\displaystyle a+bw$ is a unit if and only if $\displaystyle N(a+bw) = a^2+b^2 - ab = 1$. Thus, $\displaystyle a^2 - ab + (b^2 - 1) = 0$ we require that the discrimant, $\displaystyle b^2 - 4(b^2 - 4) > 0$, thus, $\displaystyle 3b^2 < 4 \implies b=0,1,-1$, now you can solve for $\displaystyle a$.