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.



Reduction mod $p$ of units in a ring of integers
I don't know of any result that specifically looks at units but there are a lot of results on looking at the image of a fixed finitely generated subgroup $G$ of $k^\*$ in the units of the residue fields. The granddad of these questions is the Artin conjecture for primitive roots. There are results of Gupta and Murty (which probably have been improved) and guarantee the image is everything for infinitely many primes provided that the rank of $G$ is big enough. You'd have to look at the papers to see if "big enough" applies when $G$ is group of units. If you are willing to assume GRH then much stronger results are known.


