## 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 $k \in \mathbb{Z}$.