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Suppose that $x_1, \dots, x_n$ are algebraic over $\mathbb{Q}$, and for every integer $N\geq 1$, the sum $\sum_{i=1}^{n}x_i^N$ is an algebraic integer. Does this imply that $x_1,\dots,x_n$ are also algebraic integers? If not, can we get this conclusion with additional assumption that $x_1,\dots, x_n$ and all their Galois conjugates have absolute value $1$?

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