DIVISIBILITY CONDITIONS ON THE ORDER OF THE REDUCTIONS OF ALGEBRAIC NUMBERS

Let K be a number field, and let G be a finitely generated subgroup of K^×. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes p of K such that the order of (G mod 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 for which the order is k-free, and those for which the order has a prescribed l-adic valuation for finitely many primes l. An additional condition on the Frobenius conjugacy class of 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.