Unit signature map ($\mathrm{sgn}:{\mathcal{O}}_K^{\times }\to \{\pm 1{\}}^{r_1}$)

The unit signature map records the signs of a unit under all real field embeddings. Its kernel consists of totally positive units. Its image determines the kernel of the homomorphism from the narrow ideal class group to the ordinary ideal class group: this kernel is $\{\pm 1{\}}^{r_1}/\mathrm{sgn}({\mathcal{O}}_K^{\times })$.