Let $K$ be a number field, $\rho_1,\dots,\rho_{r}$ its real embeddings, and $\sigma_1,\bar{\sigma_1},\dots,\sigma_{s},\bar{\sigma_{s}}$ its complex embeddings. There's a map \begin{align*} K &\rightarrow \prod_{i=1}^{r}\mathbb{R}\times\prod_{i=1}^{s}\mathbb{C}\\ x &\mapsto (\rho_i(x))_{i=1}^r\times(\sigma_i(x))_{i=1}^s \end{align*} If we further identify $\mathbb{C}$ with $\mathbb{R}^2$ by sending $z$ to $(\text{Re}(z),\text{Im}(z))$, then we get a map $K\rightarrow \mathbb{R}^{r+2s}$. The geometry of numbers says the ring of integers $\mathcal{O}_K$ is sent to a complete lattice in this vector space (see for example Neukirch chapter 1 proposition 5.2).
Question : Can we always choose a set of representatives of $(\mathcal{O}_K\setminus\{0\})/\mathcal{O}_K^\times$ in the vector space $\mathbb{R}^{r+2s}$ such that they all live in some cone ? Or do we have a natural choice of a set of representatives of $(\mathcal{O}_K\setminus\{0\})/\mathcal{O}_K^\times$ which admits some simple description ?
I suppose this has been already answered in the literature but don't know where to find it. Any hint or reference would be highly appreciated.
Here are some examples
- $K=\mathbb{Q}$ : Then we may choose $(\mathcal{O}_K\setminus\{0\})/\mathcal{O}_K^\times$ to be $\mathbb{N}_{\geq 1}$. So our cone is $\mathbb{R}_{\geq 0}$.
- $K=\mathbb{Q}[i]$ : Then we may choose $(\mathcal{O}_K\setminus\{0\})/\mathcal{O}_K^\times$ to be $\{a+bi\mid a\geq 1, b\geq 0\}$. So our cone is $\mathbb{R}_{\geq 0}\times \mathbb{R}_{\geq 0}$.
- $K=\mathbb{Q}[\sqrt{2}]$ : I don't know how to describe $(\mathcal{O}_K\setminus\{0\})/\mathcal{O}_K^\times$ even in this simple case.