Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$ and let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_k$. I'm looking for conditions on $k$ and $\mathfrak{p}$ which will ensure that the image of the group of units $(\mathcal{O}_k)^{\ast}$ in $\mathcal{O}_k/\mathfrak{p}$ is all of $(\mathcal{O}_k/\mathfrak{p})^{\ast}$. More generally, what is known about this image?

My area of research is geometry and not number theory (the above problem arose while studying some geometric problems), so I apologize if the answer to the above is standard stuff.