  This work concerns Artin's Conjecture on primitive roots and related problems
for number fields. Let $K$ be a number field and let $W_1$ to $W_n$ be finitely
generated subgroups of $K^\times$ of positive rank. We consider the index map,
which maps a prime $\mathfrak p$ of $K$ to the $n$-tuple of the indices of
$(W_i \bmod \mathfrak p)$. Conditionally under GRH, any preimage under the
index map admits a density, and the aim of this work is describing it. For
example, we express the density as a limit in various ways. We study in
particular the preimages of sets of $n$-tuples that are defined by prescribing
valuations for their entries. Under some mild assumptions we can express the
density as a multiple of a (suitably defined) Artin-type constant.
