The order of the reductions of an algebraic integer

Let K   be a number field, and let a∈K×a∈K×. Fix some prime number ℓ  . We compute the density of the following set: the primes pp of K such that the multiplicative order of the reduction of a   modulo pp is coprime to ℓ (or, more generally, has some prescribed ℓ-adic valuation). We evaluate the degree over K   of extensions of the form K(ζℓm,aℓn) with n≤mn≤m, which are obtained by adjoining a root of unity of order ℓmℓm and the ℓnℓn-th roots of a, as this is needed for computing the above density.