= Unit index of a CM-field
{title2=$[\mathcal O_K^\times:\mu_K\mathcal O_{K^+}^\times]\in\{1,2\}$}
For a <CM-field>, put $c$ for conjugation and $E_K=\mathcal O_K^\times$. The map $u\mapsto u/c(u)$ takes values in the cyclic group $\mu_K$ of <roots of unity>: all conjugates have absolute value one, and the <Kronecker theorem on algebraic integers in the unit disk> applies. Its kernel is $E_{K^+}$, while its image on $\mu_K$ is $\mu_K^2$. It therefore embeds $E_K/(E_{K^+}\mu_K)$ into $\mu_K/\mu_K^2$, a group of order two. This proves the index bound without assuming that every unit is a real unit times a root of unity.
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