Logarithmic embedding of number field units

Choose the real embeddings ${\sigma }_i$ and one embedding ${\tau }_j$ from each complex-conjugate pair. The weighted logarithmic embedding isFor a unit, the sum of these coordinates is $\log |N_{K/\mathbb{Q}}(u)|=0$.

The kernel of $\mathrm{Log}:{\mathcal{O}}_K^{\times }\to {\mathbb{R}}^{r+s}$ is exactly $\mu (K)$. If every conjugate of a unit has modulus one, all its powers lie in a bounded subset of the discrete Minkowski lattice ${\mathcal{O}}_K$, so two powers coincide and the unit is a root of unity.

The logarithms of a fundamental system of $r+s-1$ units form a basis of the hyperplaneand their integer span is a full lattice in $H$.

Every principal ideal $I$ of a number field has a generator $\alpha$ satisfyingat every archimedean embedding. Multiply an arbitrary generator by a unit so that its logarithmic unit-lattice coordinates lie in a fixed bounded fundamental parallelepiped.