Galois groups of order 2n that contain a cyclic subgroup of order n

Let n be any integer with n > 1, and let F ⊆ L be fields such that [L:F] = 2, L is Galois over F, and L contains a primitive nth root of unity ζ. For a cyclic Galois extension M = L(α^1/n) of L of degree n such that M is Galois over F, we determine, in terms of the action of Gal(L/F) on α and ζ, what group occurs as Gal(M/F). The general case reduces to that where n = p^e, with p prime. For n = p^e, we give an explicit parametrization of those α that lead to each possible group Gal(M/F).