<s>[INST] Associated elements in $\mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$ 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
 [/INST] 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.
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