Multiplicative Order and Frobenius Symbol for the Reductions of Number Fields

Abstract

Let L∕K be a finite Galois extension of number fields, and let G be a finitely generated subgroup of K×. We study the natural density of the set of primes of K having some prescribed Frobenius symbol in \\( \\operatorname {\\mathrm {Gal}}(L/K)\\), and for which the reduction of G has multiplicative order with some prescribed l-adic valuation for finitely many prime numbers l. This extends in several directions results by Moree and Sury (2009) and by Chinen and Tamura (2012), and has to be compared with the very general result of Ziegler (2006).