Kernel of the logarithmic unit embedding
= Kernel of the logarithmic unit embedding
The kernel of $\operatorname{Log}:\mathcal O_K^\times\to\mathbb R^{r+s}$ 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.