Let $\displaystyle

w = \frac {-1}{2} + \frac {3^{1/2}}{2}i

$, let $\displaystyle

N:Z[w] \rightarrow Z \ by\ N(a+bw) = a^2 - ab + b^2

$ and suppose that x is in Z[w]. Show that x is a unit iff N(x)=1.

My proof so far:

Assume xy = 1 for y in Z[w].

1 = N(1) = N(xy) = N(x)N(y), implies that N(x) = N(y) = 1.

Conversely, let x = a + bw with N(x) = 1.

Then N(a+bw) = 1 = a^2 - ab + b^2

Then that should equals to (a+bw)(???)

Any hints? Thanks.