 Minkowski embedding of a number field

For a number field of signature $(r_1,r_2)$, the Minkowski embedding sends an element to its values under the $r_1$ real embeddings and one representative from each of the $r_2$ complex-conjugate pairs. The image of the ring of integers is a full lattice in ${\mathbb{R}}^{r_1}\times {\mathbb{C}}^{r_2}$.