Divisibility conditions on the order of the reductions of algebraic numbers

Let K K be a number field, and let G G be a finitely generated subgroup of K × K^\times . Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes p \mathfrak p of K K such that the order of ( G mod p ) (G\bmod \mathfrak p) is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes p \mathfrak p for which the order is k k -free, and those for which the order has a prescribed ℓ \ell -adic valuation for finitely many primes ℓ \ell . An additional condition on the Frobenius conjugacy class of p \mathfrak p may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.