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

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• January 26th 2008, 12:55 PM
tttcomrader
Show Z[w] is in form of a+bw
Let $
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
• January 26th 2008, 02:01 PM
Opalg
Quote:

Originally Posted by tttcomrader
Let $
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 $w^2 = -1-w$ and $w^3=1$. Thereafter the powers of w repeat these values. It follows that any polynomial in w can be written in the form a+bw.