Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. For a given prime ideal $\mathfrak{p}$, shall we say that the map $\mathcal{O}_K^\times \to (\mathcal{O}_K/\mathfrak{p})^\times$ is surjective, where $\mathcal{O}_K^\times$ is the unit group.