Unit signature map
= Unit signature map
{title2=$\operatorname{sgn}:\mathcal O_K^\times\to\{\pm1\}^{r_1}$}
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 $\{\pm1\}^{r_1}/\operatorname{sgn}(\mathcal O_K^\times)$.