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).