Associated elements in $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$

Question

I got four elements and want to check whether these elements are associated or not. For $\alpha = \frac{1+\sqrt{-3}}{2}$ my elements are:

$a_1 = 2 - \alpha = \frac{3 - \sqrt{-3}}{2}$

$a_2 = 1 - 2\alpha = -\sqrt{-3}$

$ a_3 = 3 + 2\alpha = 4 + \sqrt{-3}$

$ a_4 = 3 - 2\alpha = 2 - \sqrt{-3}$

I just calculated $\frac{a_1}{a_2} = \frac{1-\sqrt{-3}}{2}$ and $\frac{a_2}{a_1} = \frac{-1+\sqrt{-3}}{2}$. All other pairs $\frac{a_1}{a_3}, \frac{a_1}{a_4},\frac{a_2}{a_3} \frac{a_2}{a_4}, \frac{a_3}{a_4}$ give an odd denominator which is not possble in $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ (except the denomitor 1), therefore only $a_1$ and $a_2$ are associated... is this right? Or do I miss something?

Thx for any help on this

Answer 1

I would just compute the norms first. We have $$N(a_1)=\frac{3^2+3}{4}=3,$$ $$N(a_2)=3,$$ $$N(a_3)=4^2+3=19,$$ and $$N(a_4)=2^2+3=7.$$ So the only candidates to be conjugates are $a_1$ and $a_2$, which you have verified to be the case.