Primitive roots in integer rings of number fields with class number 1

Question

In number fields that are PIDs, i.e., with class number 1, we have unique factorization of integral elements into primes, much like we do in $\Bbb Z$. Suppose $\pi$ is such a prime element in $R$, the integer ring of our number field. What can we say about the structure of $R/\pi R$?

In the case $R=\Bbb Z$, the quotient has a cyclic multiplicative group, i.e., primitive roots exist for rational primes. Is that also true in other uniquely factoring integer rings?

I know that rational primes are not necessarily prime in such rings; I’m asking about elements that are prime. Can we state necessary and sufficient conditions on an integral ring for the primitive root theorem to hold?

I have searched for an answer on MSE and Google to no avail. I have also looked at the proof that rational primes have primitive roots, but it is not clear to me whether this generalizes to integer rings in other number fields.

Answer 1

If $R$ is the ring of integers of a number field and $\mathfrak{p}$ is a non-zero prime ideal of $R$, then $R/\mathfrak{p}$ is a finite field, and so its multiplicative group is cyclic. In particular, this holds when $R$ is a PID and $\mathfrak{p}=(\pi)$, though these additional assumptions are not necessary.

Answer 2

Let $R$ be the ring of integers of a number field.

When $\pi$ is a prime element, that is $\pi$ generates a prime ideal, then $R/\pi R$ is a finite field. The multiplicative group of a finite field is cyclic; it always has a "primitive root".