4
$\begingroup$

What is the general form of elements in $\displaystyle \mathbb{Z} \left[\frac{1+\sqrt{-3}}{2} \right] $?

I'm getting muddled.

Thanks

  • 0
    Do you know what the definition of $\mathbb{Z}[\alpha]$ is (for, say, $\alpha \in \mathbb{C}$)?2011-05-22
  • 0
    I do. It's the ring of (finite) sums of powers of $ \alpha $, with integer coefficients.2011-05-22
  • 3
    If $\alpha = \frac{1+\sqrt{-3}}{2}$, can you express the square of $\alpha$ as sum of $1$ and $\alpha$ with integer coefficients ?2011-05-22
  • 0
    since $\alpha^2-\alpha+1=0$ you can write all elements as $a+b\alpha, a,b\in\mathbb{Z}$2011-05-22
  • 1
    So combine that with the fact that $\rm\ \alpha^2 \in \alpha\ \mathbb Z + \mathbb Z\ $ to conclude that $\ \mathbb Z[\alpha] \cong \ldots$2011-05-22

1 Answers 1