Let $\displaystyle
w = \frac {-1}{2} + \frac {3^{1/2}}{2}i
$
Show that each element of Z[w] can be written in the form a+bw with a,b in Z
Work out the powers of w. You should find that $\displaystyle w^2 = -1-w$ and $\displaystyle w^3=1$. Thereafter the powers of w repeat these values. It follows that any polynomial in w can be written in the form a+bw.