Describe the associates of an element in a given quadratic integer ring

Describe the associates of \sqrt{-3} in \mathbb{Q}(\sqrt{-3}) and of 2 in \mathbb{Q}(\sqrt{2}).


By Theorem 7.7 in TAN, the units in \mathbb{Q}(\sqrt{-3}) are \pm 1, (1 \pm \sqrt{-3})/2, and (-1 \pm \sqrt{-3})/2. So the associates of \sqrt{-3} are \pm \sqrt{-3} and (\pm 3 \pm \sqrt{-3})/2.

By Theorem 7.9 in TAN, the units in \mathbb{Q}(\sqrt{2}) are (1+\sqrt{2})^k with k \in \mathbb{Z}. So the associates of 2 are 2(1+\sqrt{2})^k with k \in \mathbb{Z}.

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