Logarithmic embedding of a number field

Let $K$ be a number field of signature $(r_1,r_2)$, with real embeddings ${\sigma }_1,\dots ,{\sigma }_{r_1}:K\to \mathbb{R}$ and one embedding ${\tau }_1,\dots ,{\tau }_{r_2}:K\to \mathbb{C}$ chosen from each complex conjugate pair (Archimedean embeddings and signature). The logarithmic embedding of $K$ is the map

$\lambda :K^{\times }⟶{\mathbb{R}}^{r_1+r_2},\lambda (x)=(\log \mathrm{\mid }{\sigma }_{1}x\mathrm{\mid },\dots ,\log \mathrm{\mid }{\sigma }_{r_1}x\mathrm{\mid },2\log \mathrm{\mid }{\tau }_{1}x\mathrm{\mid },\dots ,2\log \mathrm{\mid }{\tau }_{r_2}x\mathrm{\mid }),$

where $K^{\times }=K\setminus \{0\}$, $\log$ is the natural logarithm (The natural logarithm as the inverse of the exponential function), and $\mathrm{\mid }\cdot \mathrm{\mid }$ is the complex modulus (Real and imaginary parts, complex conjugation, and modulus).

The factor $2$ on the complex coordinates is part of the convention and is not optional. A real coordinate carries no factor, while a complex coordinate enters with weight $2$, matching the squared modulus $\mathrm{\mid }\tau x{\mathrm{\mid }}^2$ that is the normalized absolute value at a complex place. Taking the doubled coordinate is what makes the product-formula hyperplane the literal coordinate-sum-zero hyperplane and what fixes the determinant normalization of the regulator; for a fixed deleted row, doubling the retained complex rows multiplies the absolute determinant by $2$ for each such row. Euclidean covolumes in the corresponding hyperplanes need not change by a power of $2$.

The map is well defined. If $x\mathrm{\ne }0$ then $\sigma (x)\mathrm{\ne }0$ and $\tau (x)\mathrm{\ne }0$ for every embedding, since a field homomorphism has trivial kernel, so every modulus is a strictly positive real number and every logarithm is defined. Replacing a chosen ${\tau }_j$ by its complex conjugate does not change $\lambda$: conjugation fixes the real numbers and replaces $z=a+bi$ by $\bar{z}=a-bi$, so $\mathrm{\mid }\overline{{\tau }_{j}x}\mathrm{\mid }=\mathrm{\mid }{\tau }_{j}x\mathrm{\mid }$ for every $x\in K^{\times }$. The ordering of the coordinates is auxiliary: reordering them post-composes $\lambda$ with a linear isometry of ${\mathbb{R}}^{r_1+r_2}$, and every statement about $\lambda$ below is invariant under that reordering.

The coordinates are written in the display with the real embeddings first and the chosen complex embeddings after them; that ordering is the one used for the rest of this page.