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In this paper, we address a number of questions posed by Jones and Manes about the size of the group $G(\\phi,\\alpha) := \\text{im} \\rho = \\underset{\\leftarrow n}\\lim\\text{Gal}(K(\\phi^{-n}(\\alpha))/K)$. 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In this paper, we address a number of questions posed by Jones and Manes about the size of the group $G(\\phi,\\alpha) := \\text{im} \\rho = \\underset{\\leftarrow n}\\lim\\text{Gal}(K(\\phi^{-n}(\\alpha))/K)$. 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