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 ${\mathbb{R}}^{r_1}\times {\mathbb{C}}^{r_2}$.

This construction uses all archimedean embeddings studied in algebraic number theory.

Let $j(\mathfrak{b})$ be the Minkowski embedding of a number field applied to a nonzero fractional ideal. Under the metric $|x{|}^2={\sum }_{v\text{ real}}{x}_v^2+2{\sum }_{v\text{ complex}}|x_v{|}^2$, its covolume is $\sqrt{|D_K|}N(\mathfrak{b})$. For ordinary coordinate Lebesgue measure $\prod d{x}_v\prod d\mathrm{Re}{z}_{v}d\mathrm{Im}{z}_v$, its covolume is instead $2^{-r_2}\sqrt{|D_K|}N(\mathfrak{b})$. Here $D_K$ is the field discriminant, and the absolute norm of a fractional ideal is positive and multiplicative. The formula follows by taking the embedding determinant of an integral basis, then using the index of an integral ideal and scaling to handle a fractional ideal.