# Dirichlet’s unit theorem

Let $K$ be a number field, and let $\mathcal{O}_{K}$ be its ring of integers. Then
 $\mathcal{O}_{K}^{*}\cong\mu(K)\times\mathbb{Z}^{r+s-1}.$
Here $\mathcal{O}_{K}^{*}$ is the group of units of $\mathcal{O}_{K}$, $\mu(K)$ is the finite cyclic group of the roots of unity in $\mathcal{O}_{K}^{*}$, $r$ is the number of real embeddings $K\rightarrow\mathbb{R}$, and $2s$ is the number of non-real complex embeddings $K\rightarrow\mathbb{C}$ (which occur in complex conjugate pairs, so $s$ is an integer).