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

摘要

Let n be any integer with n > 1, and let F subset of or equal to L be fields such that [L : F] = 2, L is Galois over F, and L contains a primitive nth root of unity.. For a cyclic Galois extension M = L(alpha(1)) of L of degree n such that M is Galois over F, we determine, in terms of the action of Gal(L/F) on alpha and zeta, 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 alpha that lead to each possible group Gal(M/F). [References: 14]