# Thread: Show Z[w] is in form of a+bw

1. ## Show Z[w] is in form of a+bw

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

Let $\displaystyle w = \frac {-1}{2} + \frac {3^{1/2}}{2}i$
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.