Abstract

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 n^th 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).

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