Unscaled Minkowski embedding

Let $K$ be a number field (Number field) of degree $n=[K:\mathbb{Q}]$ and signature $(r_1,r_2)$, so that $n=r_1+2r_2$ and the field has $r_1$ real embeddings ${\sigma }_1,\dots ,{\sigma }_{r_1}:K\to \mathbb{R}$ and $r_2$ complex-conjugate pairs of nonreal embeddings from which one representative ${\tau }_1,\dots ,{\tau }_{r_2}$ is chosen; the notation and the count $r_1+2r_2=n$ are those of Archimedean embeddings and signature. The unscaled Minkowski embedding of $K$ is the injective map

$\sigma :K⟶{\mathbb{R}}^{r_1}\times {\mathbb{C}}^{r_2},\sigma (x)=({\sigma }_1(x),\dots ,{\sigma }_{r_1}(x),{\tau }_1(x),\dots ,{\tau }_{r_2}(x)),$

composed with the identification ${\mathbb{C}}^{r_2}\cong {\mathbb{R}}^{2r_2}$ that sends $z$ to the pair $(\mathrm{Re}z,\mathrm{Im}z)$ of real coordinates. This exhibits $\sigma$ as a map into ${\mathbb{R}}^{r_1}\times {\mathbb{R}}^{2r_2}={\mathbb{R}}^n$.

No factor $\sqrt{2}$ is inserted in the complex coordinates: each complex coordinate contributes the two coordinates $\mathrm{Re}{\tau }_j(x)$ and $\mathrm{Im}{\tau }_j(x)$ with equal weight. Thus the Euclidean norm of a complex block $(z_1,\dots ,z_{r_2})$ is $\sqrt{{\sum }_j\mathrm{\mid }{z}_j{\mathrm{\mid }}^2}$, and the complex block of $\sigma (x)$ has norm $\sqrt{{\sum }_j\mathrm{\mid }{\tau }_j(x){\mathrm{\mid }}^2}$. All volumes, covolumes and determinants on this item use this unscaled convention.

Injectivity and linearity. Since $n=r_1+2r_2\ge 1$, at least one of the listed embeddings exists. If $\sigma (x)=0$, every listed embedding sends $x$ to zero; any one of them is injective, so $x=0$. This also covers the case $r_2=0$, when there are no complex representatives. Each embedding is $\mathbb{Q}$-linear, as is the real-coordinate identification, so $\sigma$ is $\mathbb{Q}$-linear. Injectivity alone does not imply that the images of a $\mathbb{Q}$-basis are linearly independent over $\mathbb{R}$; that fact follows from the determinant calculation below.

Relation to the all-complex embedding determinant. Let ${\alpha }_1,\dots ,{\alpha }_n$ be a $\mathbb{Q}$-basis of $K$ and let $M=({\psi }_i({\alpha }_j))$ be the $n\times n$ matrix of all complex embeddings. By Embedding determinant formula, $\det M\mathrm{\ne }0$ and $\det (M{)}^2=\mathrm{disc}({\alpha }_1,\dots ,{\alpha }_n)$. Reorder its rows so that each complex-conjugate pair is adjacent. For a pair $\tau ,\bar{\tau }$, the old rows are obtained from the real rows $(\mathrm{Re}\tau ,\mathrm{Im}\tau )$ by the transition matrix whose determinant is $-2i$, of modulus $2$. Replacing all such pairs by their real and imaginary rows therefore gives the real $n\times n$ matrix $A$ with columns $\sigma ({\alpha }_j)$ and

$\mathrm{\mid }\det A\mathrm{\mid }=2^{-r_2}\mathrm{\mid }\det M\mathrm{\mid }=2^{-r_2}\sqrt{\mathrm{\mid }\mathrm{disc}({\alpha }_1,\dots ,{\alpha }_n)\mathrm{\mid }}\mathrm{.}$

In particular, the images of every $\mathbb{Q}$-basis form a real basis of ${\mathbb{R}}^n$. This factor $2^{-r_2}$ is responsible for the covolume formula $\mathrm{covol}(\sigma (\mathfrak{a}))=2^{-r_2}\sqrt{\mathrm{\mid }{d}_K\mathrm{\mid }}N\mathfrak{a}$ proved later in this development.