Dirichlet’s unit theorem

occur in

Let $K$ be a number field, and let ${𝒪}_K$ be its ring of integers.Then

$${𝒪}_K^{*}\cong \mu (K)\times {\mathbb{Z}}^{r+s-1}.$$

Here ${𝒪}_K^{*}$ is the group of units of ${𝒪}_K$,$\mu (K)$ is the finite cyclic group of the roots of unity in ${𝒪}_K^{*}$,$r$ is the number of real embeddings $K\to ℝ$,and $2s$ is the number of non-real complex embeddings $K\to ℂ$ (which occur in complex conjugate pairs, so $s$ is an integer).