Explicit Kummer Theory for Quadratic Fields

LetK be a quadratic number field and let α ∈ K. We present an explicit finite procedure to compute at once all Kummer degrees [K(ζm, n √ α) : K(ζm)] for n,m > 1 with n | m, where ζm denotes a primitive m-th root of unity. We can also replace α by any finitely generated subgroup of K×.